Algebra Examples

$5 x^{2} + y^{2} = 30$ $y^{2} = x^{2} + 16$

Step 1

Solve for $y$ in $y^{2} = x^{2} + 16$ .

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Step 1.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{x^{2} + 16}$

$5 x^{2} + y^{2} = 30$

Step 1.2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{x^{2} + 16}$

$5 x^{2} + y^{2} = 30$

Step 1.2.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{x^{2} + 16}$

$5 x^{2} + y^{2} = 30$

Step 1.2.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{x^{2} + 16}$

$y = - \sqrt{x^{2} + 16}$

$5 x^{2} + y^{2} = 30$

$y = \sqrt{x^{2} + 16}$

$y = - \sqrt{x^{2} + 16}$

$5 x^{2} + y^{2} = 30$

$y = \sqrt{x^{2} + 16}$

$y = - \sqrt{x^{2} + 16}$

$5 x^{2} + y^{2} = 30$

Step 2

Solve the system $y = \sqrt{x^{2} + 16}, 5 x^{2} + y^{2} = 30$ .

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Step 2.1

Replace all occurrences of $y$ with $\sqrt{x^{2} + 16}$ in each equation.

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Step 2.1.1

Replace all occurrences of $y$ in $5 x^{2} + y^{2} = 30$ with $\sqrt{x^{2} + 16}$ .

$5 x^{2} + {(\sqrt{x^{2} + 16})}^{2} = 30$

$y = \sqrt{x^{2} + 16}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $5 x^{2} + {(\sqrt{x^{2} + 16})}^{2}$ .

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Step 2.1.2.1.1

Rewrite ${\sqrt{x^{2} + 16}}^{2}$ as $x^{2} + 16$ .

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Step 2.1.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 16}$ as ${(x^{2} + 16)}^{\frac{1}{2}}$ .

$5 x^{2} + {({(x^{2} + 16)}^{\frac{1}{2}})}^{2} = 30$

$y = \sqrt{x^{2} + 16}$

Step 2.1.2.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$5 x^{2} + {(x^{2} + 16)}^{\frac{1}{2} \cdot 2} = 30$

$y = \sqrt{x^{2} + 16}$

Step 2.1.2.1.1.3

Combine $\frac{1}{2}$ and $2$ .

$5 x^{2} + {(x^{2} + 16)}^{\frac{2}{2}} = 30$

$y = \sqrt{x^{2} + 16}$

Step 2.1.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.4.1

Cancel the common factor.

$5 x^{2} + {(x^{2} + 16)}^{\frac{2}{2}} = 30$

$y = \sqrt{x^{2} + 16}$

Step 2.1.2.1.1.4.2

Rewrite the expression.

$5 x^{2} + x^{2} + 16 = 30$

$y = \sqrt{x^{2} + 16}$

$5 x^{2} + x^{2} + 16 = 30$

$y = \sqrt{x^{2} + 16}$

Step 2.1.2.1.1.5

Simplify.

$5 x^{2} + x^{2} + 16 = 30$

$y = \sqrt{x^{2} + 16}$

$5 x^{2} + x^{2} + 16 = 30$

$y = \sqrt{x^{2} + 16}$

Step 2.1.2.1.2

Add $5 x^{2}$ and $x^{2}$ .

$6 x^{2} + 16 = 30$

$y = \sqrt{x^{2} + 16}$

$6 x^{2} + 16 = 30$

$y = \sqrt{x^{2} + 16}$

$6 x^{2} + 16 = 30$

$y = \sqrt{x^{2} + 16}$

$6 x^{2} + 16 = 30$

$y = \sqrt{x^{2} + 16}$

Step 2.2

Solve for $x$ in $6 x^{2} + 16 = 30$ .

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Step 2.2.1

Move all terms not containing $x$ to the right side of the equation.

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Step 2.2.1.1

Subtract $16$ from both sides of the equation.

$6 x^{2} = 30 - 16$

$y = \sqrt{x^{2} + 16}$

Step 2.2.1.2

Subtract $16$ from $30$ .

$6 x^{2} = 14$

$y = \sqrt{x^{2} + 16}$

$6 x^{2} = 14$

$y = \sqrt{x^{2} + 16}$

Step 2.2.2

Divide each term in $6 x^{2} = 14$ by $6$ and simplify.

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Step 2.2.2.1

Divide each term in $6 x^{2} = 14$ by $6$ .

$\frac{6 x^{2}}{6} = \frac{14}{6}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $6$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{6 x^{2}}{6} = \frac{14}{6}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{14}{6}$

$y = \sqrt{x^{2} + 16}$

$x^{2} = \frac{14}{6}$

$y = \sqrt{x^{2} + 16}$

$x^{2} = \frac{14}{6}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Cancel the common factor of $14$ and $6$ .

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Step 2.2.2.3.1.1

Factor $2$ out of $14$ .

$x^{2} = \frac{2 (7)}{6}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.2.3.1.2

Cancel the common factors.

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Step 2.2.2.3.1.2.1

Factor $2$ out of $6$ .

$x^{2} = \frac{2 \cdot 7}{2 \cdot 3}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.2.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{2 \cdot 7}{2 \cdot 3}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.2.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{7}{3}$

$y = \sqrt{x^{2} + 16}$

$x^{2} = \frac{7}{3}$

$y = \sqrt{x^{2} + 16}$

$x^{2} = \frac{7}{3}$

$y = \sqrt{x^{2} + 16}$

$x^{2} = \frac{7}{3}$

$y = \sqrt{x^{2} + 16}$

$x^{2} = \frac{7}{3}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{7}{3}}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.4

Simplify $\pm \sqrt{\frac{7}{3}}$ .

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Step 2.2.4.1

Rewrite $\sqrt{\frac{7}{3}}$ as $\frac{\sqrt{7}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{7}}{\sqrt{3}}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.4.2

Multiply $\frac{\sqrt{7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{7}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.4.3

Combine and simplify the denominator.

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Step 2.2.4.3.1

Multiply $\frac{\sqrt{7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{7} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.4.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{7} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.4.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{7} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.4.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.4.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{7} \sqrt{3}}{{\sqrt{3}}^{2}}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.4.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.2.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{7} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.4.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.4.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3^{\frac{2}{2}}}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.4.3.6.4

Cancel the common factor of $2$ .

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Step 2.2.4.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3^{\frac{2}{2}}}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.4.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3}$

$y = \sqrt{x^{2} + 16}$

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.4.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3}$

$y = \sqrt{x^{2} + 16}$

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3}$

$y = \sqrt{x^{2} + 16}$

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.4.4

Simplify the numerator.

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Step 2.2.4.4.1

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{7 \cdot 3}}{3}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.4.4.2

Multiply $7$ by $3$ .

$x = \pm \frac{\sqrt{21}}{3}$

$y = \sqrt{x^{2} + 16}$

$x = \pm \frac{\sqrt{21}}{3}$

$y = \sqrt{x^{2} + 16}$

$x = \pm \frac{\sqrt{21}}{3}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{21}}{3}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{21}}{3}$

$y = \sqrt{x^{2} + 16}$

Step 2.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{21}}{3}, - \frac{\sqrt{21}}{3}$

$y = \sqrt{x^{2} + 16}$

$x = \frac{\sqrt{21}}{3}, - \frac{\sqrt{21}}{3}$

$y = \sqrt{x^{2} + 16}$

$x = \frac{\sqrt{21}}{3}, - \frac{\sqrt{21}}{3}$

$y = \sqrt{x^{2} + 16}$

Step 2.3

Replace all occurrences of $x$ with $\frac{\sqrt{21}}{3}$ in each equation.

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Step 2.3.1

Replace all occurrences of $x$ in $y = \sqrt{x^{2} + 16}$ with $\frac{\sqrt{21}}{3}$ .

$y = \sqrt{{(\frac{\sqrt{21}}{3})}^{2} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{{(\frac{\sqrt{21}}{3})}^{2} + 16}$ .

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Step 2.3.2.1.1

Apply the product rule to $\frac{\sqrt{21}}{3}$ .

$y = \sqrt{\frac{{\sqrt{21}}^{2}}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.2

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 2.3.2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$y = \sqrt{\frac{{(21^{\frac{1}{2}})}^{2}}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \sqrt{\frac{21^{\frac{1}{2} \cdot 2}}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$y = \sqrt{\frac{21^{\frac{2}{2}}}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.2.1.2.4.1

Cancel the common factor.

$y = \sqrt{\frac{21^{\frac{2}{2}}}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.2.4.2

Rewrite the expression.

$y = \sqrt{\frac{21}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

$y = \sqrt{\frac{21}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.2.5

Evaluate the exponent.

$y = \sqrt{\frac{21}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

$y = \sqrt{\frac{21}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.3

Raise $3$ to the power of $2$ .

$y = \sqrt{\frac{21}{9} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.4

Cancel the common factor of $21$ and $9$ .

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Step 2.3.2.1.4.1

Factor $3$ out of $21$ .

$y = \sqrt{\frac{3 (7)}{9} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.4.2

Cancel the common factors.

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Step 2.3.2.1.4.2.1

Factor $3$ out of $9$ .

$y = \sqrt{\frac{3 \cdot 7}{3 \cdot 3} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.4.2.2

Cancel the common factor.

$y = \sqrt{\frac{3 \cdot 7}{3 \cdot 3} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.4.2.3

Rewrite the expression.

$y = \sqrt{\frac{7}{3} + 16}$

$x = \frac{\sqrt{21}}{3}$

$y = \sqrt{\frac{7}{3} + 16}$

$x = \frac{\sqrt{21}}{3}$

$y = \sqrt{\frac{7}{3} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.5

To write $16$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \sqrt{\frac{7}{3} + 16 \cdot \frac{3}{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.6

Combine $16$ and $\frac{3}{3}$ .

$y = \sqrt{\frac{7}{3} + \frac{16 \cdot 3}{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.7

Combine the numerators over the common denominator.

$y = \sqrt{\frac{7 + 16 \cdot 3}{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.8

Simplify the numerator.

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Step 2.3.2.1.8.1

Multiply $16$ by $3$ .

$y = \sqrt{\frac{7 + 48}{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.8.2

Add $7$ and $48$ .

$y = \sqrt{\frac{55}{3}}$

$x = \frac{\sqrt{21}}{3}$

$y = \sqrt{\frac{55}{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.9

Rewrite $\sqrt{\frac{55}{3}}$ as $\frac{\sqrt{55}}{\sqrt{3}}$ .

$y = \frac{\sqrt{55}}{\sqrt{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.10

Multiply $\frac{\sqrt{55}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \frac{\sqrt{55}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.11

Combine and simplify the denominator.

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Step 2.3.2.1.11.1

Multiply $\frac{\sqrt{55}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \frac{\sqrt{55} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.11.2

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{\sqrt{55} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.11.3

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{\sqrt{55} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.11.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{\sqrt{55} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.11.5

Add $1$ and $1$ .

$y = \frac{\sqrt{55} \sqrt{3}}{{\sqrt{3}}^{2}}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.11.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.3.2.1.11.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = \frac{\sqrt{55} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.11.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{\sqrt{55} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.11.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \frac{\sqrt{55} \sqrt{3}}{3^{\frac{2}{2}}}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.11.6.4

Cancel the common factor of $2$ .

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Step 2.3.2.1.11.6.4.1

Cancel the common factor.

$y = \frac{\sqrt{55} \sqrt{3}}{3^{\frac{2}{2}}}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.11.6.4.2

Rewrite the expression.

$y = \frac{\sqrt{55} \sqrt{3}}{3}$

$x = \frac{\sqrt{21}}{3}$

$y = \frac{\sqrt{55} \sqrt{3}}{3}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.11.6.5

Evaluate the exponent.

$y = \frac{\sqrt{55} \sqrt{3}}{3}$

$x = \frac{\sqrt{21}}{3}$

$y = \frac{\sqrt{55} \sqrt{3}}{3}$

$x = \frac{\sqrt{21}}{3}$

$y = \frac{\sqrt{55} \sqrt{3}}{3}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.12

Simplify the numerator.

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Step 2.3.2.1.12.1

Combine using the product rule for radicals.

$y = \frac{\sqrt{55 \cdot 3}}{3}$

$x = \frac{\sqrt{21}}{3}$

Step 2.3.2.1.12.2

Multiply $55$ by $3$ .

$y = \frac{\sqrt{165}}{3}$

$x = \frac{\sqrt{21}}{3}$

$y = \frac{\sqrt{165}}{3}$

$x = \frac{\sqrt{21}}{3}$

$y = \frac{\sqrt{165}}{3}$

$x = \frac{\sqrt{21}}{3}$

$y = \frac{\sqrt{165}}{3}$

$x = \frac{\sqrt{21}}{3}$

$y = \frac{\sqrt{165}}{3}$

$x = \frac{\sqrt{21}}{3}$

Step 2.4

Replace all occurrences of $x$ with $- \frac{\sqrt{21}}{3}$ in each equation.

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Step 2.4.1

Replace all occurrences of $x$ in $y = \sqrt{x^{2} + 16}$ with $- \frac{\sqrt{21}}{3}$ .

$y = \sqrt{{(- \frac{\sqrt{21}}{3})}^{2} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\sqrt{{(- \frac{\sqrt{21}}{3})}^{2} + 16}$ .

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Step 2.4.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.4.2.1.1.1

Apply the product rule to $- \frac{\sqrt{21}}{3}$ .

$y = \sqrt{{(- 1)}^{2} {(\frac{\sqrt{21}}{3})}^{2} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.1.2

Apply the product rule to $\frac{\sqrt{21}}{3}$ .

$y = \sqrt{{(- 1)}^{2} (\frac{{\sqrt{21}}^{2}}{3^{2}}) + 16}$

$x = - \frac{\sqrt{21}}{3}$

$y = \sqrt{{(- 1)}^{2} (\frac{{\sqrt{21}}^{2}}{3^{2}}) + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.2

Simplify the expression.

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Step 2.4.2.1.2.1

Raise $- 1$ to the power of $2$ .

$y = \sqrt{1 (\frac{{\sqrt{21}}^{2}}{3^{2}}) + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.2.2

Multiply $\frac{{\sqrt{21}}^{2}}{3^{2}}$ by $1$ .

$y = \sqrt{\frac{{\sqrt{21}}^{2}}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

$y = \sqrt{\frac{{\sqrt{21}}^{2}}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.3

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 2.4.2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$y = \sqrt{\frac{{(21^{\frac{1}{2}})}^{2}}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \sqrt{\frac{21^{\frac{1}{2} \cdot 2}}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.3.3

Combine $\frac{1}{2}$ and $2$ .

$y = \sqrt{\frac{21^{\frac{2}{2}}}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.3.4

Cancel the common factor of $2$ .

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Step 2.4.2.1.3.4.1

Cancel the common factor.

$y = \sqrt{\frac{21^{\frac{2}{2}}}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.3.4.2

Rewrite the expression.

$y = \sqrt{\frac{21}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

$y = \sqrt{\frac{21}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.3.5

Evaluate the exponent.

$y = \sqrt{\frac{21}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

$y = \sqrt{\frac{21}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.4

Raise $3$ to the power of $2$ .

$y = \sqrt{\frac{21}{9} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.5

Cancel the common factor of $21$ and $9$ .

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Step 2.4.2.1.5.1

Factor $3$ out of $21$ .

$y = \sqrt{\frac{3 (7)}{9} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.5.2

Cancel the common factors.

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Step 2.4.2.1.5.2.1

Factor $3$ out of $9$ .

$y = \sqrt{\frac{3 \cdot 7}{3 \cdot 3} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.5.2.2

Cancel the common factor.

$y = \sqrt{\frac{3 \cdot 7}{3 \cdot 3} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.5.2.3

Rewrite the expression.

$y = \sqrt{\frac{7}{3} + 16}$

$x = - \frac{\sqrt{21}}{3}$

$y = \sqrt{\frac{7}{3} + 16}$

$x = - \frac{\sqrt{21}}{3}$

$y = \sqrt{\frac{7}{3} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.6

To write $16$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \sqrt{\frac{7}{3} + 16 \cdot \frac{3}{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.7

Combine $16$ and $\frac{3}{3}$ .

$y = \sqrt{\frac{7}{3} + \frac{16 \cdot 3}{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.8

Combine the numerators over the common denominator.

$y = \sqrt{\frac{7 + 16 \cdot 3}{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.9

Simplify the numerator.

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Step 2.4.2.1.9.1

Multiply $16$ by $3$ .

$y = \sqrt{\frac{7 + 48}{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.9.2

Add $7$ and $48$ .

$y = \sqrt{\frac{55}{3}}$

$x = - \frac{\sqrt{21}}{3}$

$y = \sqrt{\frac{55}{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.10

Rewrite $\sqrt{\frac{55}{3}}$ as $\frac{\sqrt{55}}{\sqrt{3}}$ .

$y = \frac{\sqrt{55}}{\sqrt{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.11

Multiply $\frac{\sqrt{55}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \frac{\sqrt{55}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.12

Combine and simplify the denominator.

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Step 2.4.2.1.12.1

Multiply $\frac{\sqrt{55}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \frac{\sqrt{55} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.12.2

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{\sqrt{55} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.12.3

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{\sqrt{55} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.12.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{\sqrt{55} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.12.5

Add $1$ and $1$ .

$y = \frac{\sqrt{55} \sqrt{3}}{{\sqrt{3}}^{2}}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.12.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.4.2.1.12.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = \frac{\sqrt{55} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.12.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{\sqrt{55} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.12.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \frac{\sqrt{55} \sqrt{3}}{3^{\frac{2}{2}}}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.12.6.4

Cancel the common factor of $2$ .

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Step 2.4.2.1.12.6.4.1

Cancel the common factor.

$y = \frac{\sqrt{55} \sqrt{3}}{3^{\frac{2}{2}}}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.12.6.4.2

Rewrite the expression.

$y = \frac{\sqrt{55} \sqrt{3}}{3}$

$x = - \frac{\sqrt{21}}{3}$

$y = \frac{\sqrt{55} \sqrt{3}}{3}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.12.6.5

Evaluate the exponent.

$y = \frac{\sqrt{55} \sqrt{3}}{3}$

$x = - \frac{\sqrt{21}}{3}$

$y = \frac{\sqrt{55} \sqrt{3}}{3}$

$x = - \frac{\sqrt{21}}{3}$

$y = \frac{\sqrt{55} \sqrt{3}}{3}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.13

Simplify the numerator.

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Step 2.4.2.1.13.1

Combine using the product rule for radicals.

$y = \frac{\sqrt{55 \cdot 3}}{3}$

$x = - \frac{\sqrt{21}}{3}$

Step 2.4.2.1.13.2

Multiply $55$ by $3$ .

$y = \frac{\sqrt{165}}{3}$

$x = - \frac{\sqrt{21}}{3}$

$y = \frac{\sqrt{165}}{3}$

$x = - \frac{\sqrt{21}}{3}$

$y = \frac{\sqrt{165}}{3}$

$x = - \frac{\sqrt{21}}{3}$

$y = \frac{\sqrt{165}}{3}$

$x = - \frac{\sqrt{21}}{3}$

$y = \frac{\sqrt{165}}{3}$

$x = - \frac{\sqrt{21}}{3}$

$y = \frac{\sqrt{165}}{3}$

$x = - \frac{\sqrt{21}}{3}$

Step 3

Solve the system $y = - \sqrt{x^{2} + 16}, 5 x^{2} + y^{2} = 30$ .

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Step 3.1

Replace all occurrences of $y$ with $- \sqrt{x^{2} + 16}$ in each equation.

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Step 3.1.1

Replace all occurrences of $y$ in $5 x^{2} + y^{2} = 30$ with $- \sqrt{x^{2} + 16}$ .

$5 x^{2} + {(- \sqrt{x^{2} + 16})}^{2} = 30$

$y = - \sqrt{x^{2} + 16}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify $5 x^{2} + {(- \sqrt{x^{2} + 16})}^{2}$ .

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Step 3.1.2.1.1

Simplify each term.

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Step 3.1.2.1.1.1

Apply the product rule to $- \sqrt{x^{2} + 16}$ .

$5 x^{2} + {(- 1)}^{2} {\sqrt{x^{2} + 16}}^{2} = 30$

$y = - \sqrt{x^{2} + 16}$

Step 3.1.2.1.1.2

Raise $- 1$ to the power of $2$ .

$5 x^{2} + 1 {\sqrt{x^{2} + 16}}^{2} = 30$

$y = - \sqrt{x^{2} + 16}$

Step 3.1.2.1.1.3

Multiply ${\sqrt{x^{2} + 16}}^{2}$ by $1$ .

$5 x^{2} + {\sqrt{x^{2} + 16}}^{2} = 30$

$y = - \sqrt{x^{2} + 16}$

Step 3.1.2.1.1.4

Rewrite ${\sqrt{x^{2} + 16}}^{2}$ as $x^{2} + 16$ .

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Step 3.1.2.1.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 16}$ as ${(x^{2} + 16)}^{\frac{1}{2}}$ .

$5 x^{2} + {({(x^{2} + 16)}^{\frac{1}{2}})}^{2} = 30$

$y = - \sqrt{x^{2} + 16}$

Step 3.1.2.1.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$5 x^{2} + {(x^{2} + 16)}^{\frac{1}{2} \cdot 2} = 30$

$y = - \sqrt{x^{2} + 16}$

Step 3.1.2.1.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$5 x^{2} + {(x^{2} + 16)}^{\frac{2}{2}} = 30$

$y = - \sqrt{x^{2} + 16}$

Step 3.1.2.1.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.1.4.4.1

Cancel the common factor.

$5 x^{2} + {(x^{2} + 16)}^{\frac{2}{2}} = 30$

$y = - \sqrt{x^{2} + 16}$

Step 3.1.2.1.1.4.4.2

Rewrite the expression.

$5 x^{2} + x^{2} + 16 = 30$

$y = - \sqrt{x^{2} + 16}$

$5 x^{2} + x^{2} + 16 = 30$

$y = - \sqrt{x^{2} + 16}$

Step 3.1.2.1.1.4.5

Simplify.

$5 x^{2} + x^{2} + 16 = 30$

$y = - \sqrt{x^{2} + 16}$

$5 x^{2} + x^{2} + 16 = 30$

$y = - \sqrt{x^{2} + 16}$

$5 x^{2} + x^{2} + 16 = 30$

$y = - \sqrt{x^{2} + 16}$

Step 3.1.2.1.2

Add $5 x^{2}$ and $x^{2}$ .

$6 x^{2} + 16 = 30$

$y = - \sqrt{x^{2} + 16}$

$6 x^{2} + 16 = 30$

$y = - \sqrt{x^{2} + 16}$

$6 x^{2} + 16 = 30$

$y = - \sqrt{x^{2} + 16}$

$6 x^{2} + 16 = 30$

$y = - \sqrt{x^{2} + 16}$

Step 3.2

Solve for $x$ in $6 x^{2} + 16 = 30$ .

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Step 3.2.1

Move all terms not containing $x$ to the right side of the equation.

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Step 3.2.1.1

Subtract $16$ from both sides of the equation.

$6 x^{2} = 30 - 16$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.1.2

Subtract $16$ from $30$ .

$6 x^{2} = 14$

$y = - \sqrt{x^{2} + 16}$

$6 x^{2} = 14$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.2

Divide each term in $6 x^{2} = 14$ by $6$ and simplify.

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Step 3.2.2.1

Divide each term in $6 x^{2} = 14$ by $6$ .

$\frac{6 x^{2}}{6} = \frac{14}{6}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $6$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{6 x^{2}}{6} = \frac{14}{6}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{14}{6}$

$y = - \sqrt{x^{2} + 16}$

$x^{2} = \frac{14}{6}$

$y = - \sqrt{x^{2} + 16}$

$x^{2} = \frac{14}{6}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Cancel the common factor of $14$ and $6$ .

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Step 3.2.2.3.1.1

Factor $2$ out of $14$ .

$x^{2} = \frac{2 (7)}{6}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.2.3.1.2

Cancel the common factors.

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Step 3.2.2.3.1.2.1

Factor $2$ out of $6$ .

$x^{2} = \frac{2 \cdot 7}{2 \cdot 3}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.2.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{2 \cdot 7}{2 \cdot 3}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.2.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{7}{3}$

$y = - \sqrt{x^{2} + 16}$

$x^{2} = \frac{7}{3}$

$y = - \sqrt{x^{2} + 16}$

$x^{2} = \frac{7}{3}$

$y = - \sqrt{x^{2} + 16}$

$x^{2} = \frac{7}{3}$

$y = - \sqrt{x^{2} + 16}$

$x^{2} = \frac{7}{3}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{7}{3}}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.4

Simplify $\pm \sqrt{\frac{7}{3}}$ .

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Step 3.2.4.1

Rewrite $\sqrt{\frac{7}{3}}$ as $\frac{\sqrt{7}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{7}}{\sqrt{3}}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.4.2

Multiply $\frac{\sqrt{7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{7}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.4.3

Combine and simplify the denominator.

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Step 3.2.4.3.1

Multiply $\frac{\sqrt{7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{7} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.4.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{7} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.4.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{7} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.4.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.4.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{7} \sqrt{3}}{{\sqrt{3}}^{2}}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.4.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.2.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{7} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.4.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.4.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3^{\frac{2}{2}}}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.4.3.6.4

Cancel the common factor of $2$ .

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Step 3.2.4.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3^{\frac{2}{2}}}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.4.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3}$

$y = - \sqrt{x^{2} + 16}$

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.4.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3}$

$y = - \sqrt{x^{2} + 16}$

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3}$

$y = - \sqrt{x^{2} + 16}$

$x = \pm \frac{\sqrt{7} \sqrt{3}}{3}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.4.4

Simplify the numerator.

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Step 3.2.4.4.1

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{7 \cdot 3}}{3}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.4.4.2

Multiply $7$ by $3$ .

$x = \pm \frac{\sqrt{21}}{3}$

$y = - \sqrt{x^{2} + 16}$

$x = \pm \frac{\sqrt{21}}{3}$

$y = - \sqrt{x^{2} + 16}$

$x = \pm \frac{\sqrt{21}}{3}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{21}}{3}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{21}}{3}$

$y = - \sqrt{x^{2} + 16}$

Step 3.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{21}}{3}, - \frac{\sqrt{21}}{3}$

$y = - \sqrt{x^{2} + 16}$

$x = \frac{\sqrt{21}}{3}, - \frac{\sqrt{21}}{3}$

$y = - \sqrt{x^{2} + 16}$

$x = \frac{\sqrt{21}}{3}, - \frac{\sqrt{21}}{3}$

$y = - \sqrt{x^{2} + 16}$

Step 3.3

Replace all occurrences of $x$ with $\frac{\sqrt{21}}{3}$ in each equation.

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Step 3.3.1

Replace all occurrences of $x$ in $y = - \sqrt{x^{2} + 16}$ with $\frac{\sqrt{21}}{3}$ .

$y = - \sqrt{{(\frac{\sqrt{21}}{3})}^{2} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \sqrt{{(\frac{\sqrt{21}}{3})}^{2} + 16}$ .

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Step 3.3.2.1.1

Apply the product rule to $\frac{\sqrt{21}}{3}$ .

$y = - \sqrt{\frac{{\sqrt{21}}^{2}}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.2

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 3.3.2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$y = - \sqrt{\frac{{(21^{\frac{1}{2}})}^{2}}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = - \sqrt{\frac{21^{\frac{1}{2} \cdot 2}}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$y = - \sqrt{\frac{21^{\frac{2}{2}}}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.2.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.2.4.1

Cancel the common factor.

$y = - \sqrt{\frac{21^{\frac{2}{2}}}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.2.4.2

Rewrite the expression.

$y = - \sqrt{\frac{21}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

$y = - \sqrt{\frac{21}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.2.5

Evaluate the exponent.

$y = - \sqrt{\frac{21}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

$y = - \sqrt{\frac{21}{3^{2}} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.3

Raise $3$ to the power of $2$ .

$y = - \sqrt{\frac{21}{9} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.4

Cancel the common factor of $21$ and $9$ .

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Step 3.3.2.1.4.1

Factor $3$ out of $21$ .

$y = - \sqrt{\frac{3 (7)}{9} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.4.2

Cancel the common factors.

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Step 3.3.2.1.4.2.1

Factor $3$ out of $9$ .

$y = - \sqrt{\frac{3 \cdot 7}{3 \cdot 3} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.4.2.2

Cancel the common factor.

$y = - \sqrt{\frac{3 \cdot 7}{3 \cdot 3} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.4.2.3

Rewrite the expression.

$y = - \sqrt{\frac{7}{3} + 16}$

$x = \frac{\sqrt{21}}{3}$

$y = - \sqrt{\frac{7}{3} + 16}$

$x = \frac{\sqrt{21}}{3}$

$y = - \sqrt{\frac{7}{3} + 16}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.5

To write $16$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \sqrt{\frac{7}{3} + 16 \cdot \frac{3}{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.6

Combine $16$ and $\frac{3}{3}$ .

$y = - \sqrt{\frac{7}{3} + \frac{16 \cdot 3}{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.7

Combine the numerators over the common denominator.

$y = - \sqrt{\frac{7 + 16 \cdot 3}{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.8

Simplify the numerator.

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Step 3.3.2.1.8.1

Multiply $16$ by $3$ .

$y = - \sqrt{\frac{7 + 48}{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.8.2

Add $7$ and $48$ .

$y = - \sqrt{\frac{55}{3}}$

$x = \frac{\sqrt{21}}{3}$

$y = - \sqrt{\frac{55}{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.9

Rewrite $\sqrt{\frac{55}{3}}$ as $\frac{\sqrt{55}}{\sqrt{3}}$ .

$y = - \frac{\sqrt{55}}{\sqrt{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.10

Multiply $\frac{\sqrt{55}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = - (\frac{\sqrt{55}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.11

Combine and simplify the denominator.

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Step 3.3.2.1.11.1

Multiply $\frac{\sqrt{55}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = - \frac{\sqrt{55} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.11.2

Raise $\sqrt{3}$ to the power of $1$ .

$y = - \frac{\sqrt{55} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.11.3

Raise $\sqrt{3}$ to the power of $1$ .

$y = - \frac{\sqrt{55} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.11.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = - \frac{\sqrt{55} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.11.5

Add $1$ and $1$ .

$y = - \frac{\sqrt{55} \sqrt{3}}{{\sqrt{3}}^{2}}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.11.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.3.2.1.11.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = - \frac{\sqrt{55} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.11.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = - \frac{\sqrt{55} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.11.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = - \frac{\sqrt{55} \sqrt{3}}{3^{\frac{2}{2}}}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.11.6.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.11.6.4.1

Cancel the common factor.

$y = - \frac{\sqrt{55} \sqrt{3}}{3^{\frac{2}{2}}}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.11.6.4.2

Rewrite the expression.

$y = - \frac{\sqrt{55} \sqrt{3}}{3}$

$x = \frac{\sqrt{21}}{3}$

$y = - \frac{\sqrt{55} \sqrt{3}}{3}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.11.6.5

Evaluate the exponent.

$y = - \frac{\sqrt{55} \sqrt{3}}{3}$

$x = \frac{\sqrt{21}}{3}$

$y = - \frac{\sqrt{55} \sqrt{3}}{3}$

$x = \frac{\sqrt{21}}{3}$

$y = - \frac{\sqrt{55} \sqrt{3}}{3}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.12

Simplify the numerator.

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Step 3.3.2.1.12.1

Combine using the product rule for radicals.

$y = - \frac{\sqrt{55 \cdot 3}}{3}$

$x = \frac{\sqrt{21}}{3}$

Step 3.3.2.1.12.2

Multiply $55$ by $3$ .

$y = - \frac{\sqrt{165}}{3}$

$x = \frac{\sqrt{21}}{3}$

$y = - \frac{\sqrt{165}}{3}$

$x = \frac{\sqrt{21}}{3}$

$y = - \frac{\sqrt{165}}{3}$

$x = \frac{\sqrt{21}}{3}$

$y = - \frac{\sqrt{165}}{3}$

$x = \frac{\sqrt{21}}{3}$

$y = - \frac{\sqrt{165}}{3}$

$x = \frac{\sqrt{21}}{3}$

Step 3.4

Replace all occurrences of $x$ with $- \frac{\sqrt{21}}{3}$ in each equation.

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Step 3.4.1

Replace all occurrences of $x$ in $y = - \sqrt{x^{2} + 16}$ with $- \frac{\sqrt{21}}{3}$ .

$y = - \sqrt{{(- \frac{\sqrt{21}}{3})}^{2} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- \sqrt{{(- \frac{\sqrt{21}}{3})}^{2} + 16}$ .

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Step 3.4.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.4.2.1.1.1

Apply the product rule to $- \frac{\sqrt{21}}{3}$ .

$y = - \sqrt{{(- 1)}^{2} {(\frac{\sqrt{21}}{3})}^{2} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.1.2

Apply the product rule to $\frac{\sqrt{21}}{3}$ .

$y = - \sqrt{{(- 1)}^{2} (\frac{{\sqrt{21}}^{2}}{3^{2}}) + 16}$

$x = - \frac{\sqrt{21}}{3}$

$y = - \sqrt{{(- 1)}^{2} (\frac{{\sqrt{21}}^{2}}{3^{2}}) + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.2

Simplify the expression.

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Step 3.4.2.1.2.1

Raise $- 1$ to the power of $2$ .

$y = - \sqrt{1 (\frac{{\sqrt{21}}^{2}}{3^{2}}) + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.2.2

Multiply $\frac{{\sqrt{21}}^{2}}{3^{2}}$ by $1$ .

$y = - \sqrt{\frac{{\sqrt{21}}^{2}}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

$y = - \sqrt{\frac{{\sqrt{21}}^{2}}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.3

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 3.4.2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$y = - \sqrt{\frac{{(21^{\frac{1}{2}})}^{2}}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = - \sqrt{\frac{21^{\frac{1}{2} \cdot 2}}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.3.3

Combine $\frac{1}{2}$ and $2$ .

$y = - \sqrt{\frac{21^{\frac{2}{2}}}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.3.4

Cancel the common factor of $2$ .

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Step 3.4.2.1.3.4.1

Cancel the common factor.

$y = - \sqrt{\frac{21^{\frac{2}{2}}}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.3.4.2

Rewrite the expression.

$y = - \sqrt{\frac{21}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

$y = - \sqrt{\frac{21}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.3.5

Evaluate the exponent.

$y = - \sqrt{\frac{21}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

$y = - \sqrt{\frac{21}{3^{2}} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.4

Raise $3$ to the power of $2$ .

$y = - \sqrt{\frac{21}{9} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.5

Cancel the common factor of $21$ and $9$ .

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Step 3.4.2.1.5.1

Factor $3$ out of $21$ .

$y = - \sqrt{\frac{3 (7)}{9} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.5.2

Cancel the common factors.

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Step 3.4.2.1.5.2.1

Factor $3$ out of $9$ .

$y = - \sqrt{\frac{3 \cdot 7}{3 \cdot 3} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.5.2.2

Cancel the common factor.

$y = - \sqrt{\frac{3 \cdot 7}{3 \cdot 3} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.5.2.3

Rewrite the expression.

$y = - \sqrt{\frac{7}{3} + 16}$

$x = - \frac{\sqrt{21}}{3}$

$y = - \sqrt{\frac{7}{3} + 16}$

$x = - \frac{\sqrt{21}}{3}$

$y = - \sqrt{\frac{7}{3} + 16}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.6

To write $16$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \sqrt{\frac{7}{3} + 16 \cdot \frac{3}{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.7

Combine $16$ and $\frac{3}{3}$ .

$y = - \sqrt{\frac{7}{3} + \frac{16 \cdot 3}{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.8

Combine the numerators over the common denominator.

$y = - \sqrt{\frac{7 + 16 \cdot 3}{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.9

Simplify the numerator.

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Step 3.4.2.1.9.1

Multiply $16$ by $3$ .

$y = - \sqrt{\frac{7 + 48}{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.9.2

Add $7$ and $48$ .

$y = - \sqrt{\frac{55}{3}}$

$x = - \frac{\sqrt{21}}{3}$

$y = - \sqrt{\frac{55}{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.10

Rewrite $\sqrt{\frac{55}{3}}$ as $\frac{\sqrt{55}}{\sqrt{3}}$ .

$y = - \frac{\sqrt{55}}{\sqrt{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.11

Multiply $\frac{\sqrt{55}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = - (\frac{\sqrt{55}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.12

Combine and simplify the denominator.

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Step 3.4.2.1.12.1

Multiply $\frac{\sqrt{55}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = - \frac{\sqrt{55} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.12.2

Raise $\sqrt{3}$ to the power of $1$ .

$y = - \frac{\sqrt{55} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.12.3

Raise $\sqrt{3}$ to the power of $1$ .

$y = - \frac{\sqrt{55} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.12.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = - \frac{\sqrt{55} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.12.5

Add $1$ and $1$ .

$y = - \frac{\sqrt{55} \sqrt{3}}{{\sqrt{3}}^{2}}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.12.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.4.2.1.12.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = - \frac{\sqrt{55} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.12.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = - \frac{\sqrt{55} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.12.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = - \frac{\sqrt{55} \sqrt{3}}{3^{\frac{2}{2}}}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.12.6.4

Cancel the common factor of $2$ .

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Step 3.4.2.1.12.6.4.1

Cancel the common factor.

$y = - \frac{\sqrt{55} \sqrt{3}}{3^{\frac{2}{2}}}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.12.6.4.2

Rewrite the expression.

$y = - \frac{\sqrt{55} \sqrt{3}}{3}$

$x = - \frac{\sqrt{21}}{3}$

$y = - \frac{\sqrt{55} \sqrt{3}}{3}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.12.6.5

Evaluate the exponent.

$y = - \frac{\sqrt{55} \sqrt{3}}{3}$

$x = - \frac{\sqrt{21}}{3}$

$y = - \frac{\sqrt{55} \sqrt{3}}{3}$

$x = - \frac{\sqrt{21}}{3}$

$y = - \frac{\sqrt{55} \sqrt{3}}{3}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.13

Simplify the numerator.

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Step 3.4.2.1.13.1

Combine using the product rule for radicals.

$y = - \frac{\sqrt{55 \cdot 3}}{3}$

$x = - \frac{\sqrt{21}}{3}$

Step 3.4.2.1.13.2

Multiply $55$ by $3$ .

$y = - \frac{\sqrt{165}}{3}$

$x = - \frac{\sqrt{21}}{3}$

$y = - \frac{\sqrt{165}}{3}$

$x = - \frac{\sqrt{21}}{3}$

$y = - \frac{\sqrt{165}}{3}$

$x = - \frac{\sqrt{21}}{3}$

$y = - \frac{\sqrt{165}}{3}$

$x = - \frac{\sqrt{21}}{3}$

$y = - \frac{\sqrt{165}}{3}$

$x = - \frac{\sqrt{21}}{3}$

$y = - \frac{\sqrt{165}}{3}$

$x = - \frac{\sqrt{21}}{3}$

Step 4

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(\frac{\sqrt{21}}{3}, \frac{\sqrt{165}}{3})$

$(- \frac{\sqrt{21}}{3}, \frac{\sqrt{165}}{3})$

$(\frac{\sqrt{21}}{3}, - \frac{\sqrt{165}}{3})$

$(- \frac{\sqrt{21}}{3}, - \frac{\sqrt{165}}{3})$

Step 5

The result can be shown in multiple forms.

Point Form:

$(\frac{\sqrt{21}}{3}, \frac{\sqrt{165}}{3}), (- \frac{\sqrt{21}}{3}, \frac{\sqrt{165}}{3}), (\frac{\sqrt{21}}{3}, - \frac{\sqrt{165}}{3}), (- \frac{\sqrt{21}}{3}, - \frac{\sqrt{165}}{3})$

Equation Form:

$x = \frac{\sqrt{21}}{3}, y = \frac{\sqrt{165}}{3}$

$x = - \frac{\sqrt{21}}{3}, y = \frac{\sqrt{165}}{3}$

$x = \frac{\sqrt{21}}{3}, y = - \frac{\sqrt{165}}{3}$

$x = - \frac{\sqrt{21}}{3}, y = - \frac{\sqrt{165}}{3}$

Step 6