Algebra Examples

$\sqrt{{(\frac{\sqrt{11}}{11})}^{2} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 1

Apply the product rule to $\frac{\sqrt{11}}{11}$ .

$\sqrt{\frac{{\sqrt{11}}^{2}}{11^{2}} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 2

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\sqrt{\frac{{(11^{\frac{1}{2}})}^{2}}{11^{2}} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 2.2

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

$\sqrt{\frac{11^{\frac{1}{2} \cdot 2}}{11^{2}} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 2.3

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

$\sqrt{\frac{11^{\frac{2}{2}}}{11^{2}} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

$\sqrt{\frac{11^{\frac{2}{2}}}{11^{2}} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 2.4.2

Rewrite the expression.

$\sqrt{\frac{11^{1}}{11^{2}} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 2.5

Evaluate the exponent.

$\sqrt{\frac{11}{11^{2}} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 3

Raise $11$ to the power of $2$ .

$\sqrt{\frac{11}{121} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 4

Cancel the common factor of $11$ and $121$ .

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Step 4.1

Factor $11$ out of $11$ .

$\sqrt{\frac{11 (1)}{121} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $11$ out of $121$ .

$\sqrt{\frac{11 \cdot 1}{11 \cdot 11} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 4.2.2

Cancel the common factor.

$\sqrt{\frac{11 \cdot 1}{11 \cdot 11} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 4.2.3

Rewrite the expression.

$\sqrt{\frac{1}{11} + {(\frac{\sqrt{11}}{11})}^{2} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 5

Apply the product rule to $\frac{\sqrt{11}}{11}$ .

$\sqrt{\frac{1}{11} + \frac{{\sqrt{11}}^{2}}{11^{2}} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 6

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\sqrt{\frac{1}{11} + \frac{{(11^{\frac{1}{2}})}^{2}}{11^{2}} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 6.2

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

$\sqrt{\frac{1}{11} + \frac{11^{\frac{1}{2} \cdot 2}}{11^{2}} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 6.3

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

$\sqrt{\frac{1}{11} + \frac{11^{\frac{2}{2}}}{11^{2}} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Cancel the common factor.

$\sqrt{\frac{1}{11} + \frac{11^{\frac{2}{2}}}{11^{2}} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 6.4.2

Rewrite the expression.

$\sqrt{\frac{1}{11} + \frac{11^{1}}{11^{2}} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 6.5

Evaluate the exponent.

$\sqrt{\frac{1}{11} + \frac{11}{11^{2}} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 7

Raise $11$ to the power of $2$ .

$\sqrt{\frac{1}{11} + \frac{11}{121} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 8

Cancel the common factor of $11$ and $121$ .

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Step 8.1

Factor $11$ out of $11$ .

$\sqrt{\frac{1}{11} + \frac{11 (1)}{121} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $11$ out of $121$ .

$\sqrt{\frac{1}{11} + \frac{11 \cdot 1}{11 \cdot 11} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 8.2.2

Cancel the common factor.

$\sqrt{\frac{1}{11} + \frac{11 \cdot 1}{11 \cdot 11} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 8.2.3

Rewrite the expression.

$\sqrt{\frac{1}{11} + \frac{1}{11} + {(\frac{3 \sqrt{11}}{11})}^{2}}$

Step 9

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

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Step 9.1

Apply the product rule to $\frac{3 \sqrt{11}}{11}$ .

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{{(3 \sqrt{11})}^{2}}{11^{2}}}$

Step 9.2

Apply the product rule to $3 \sqrt{11}$ .

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{3^{2} {\sqrt{11}}^{2}}{11^{2}}}$

Step 10

Simplify the numerator.

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Step 10.1

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

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9 {\sqrt{11}}^{2}}{11^{2}}}$

Step 10.2

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 10.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9 {(11^{\frac{1}{2}})}^{2}}{11^{2}}}$

Step 10.2.2

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

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9 \cdot 11^{\frac{1}{2} \cdot 2}}{11^{2}}}$

Step 10.2.3

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

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9 \cdot 11^{\frac{2}{2}}}{11^{2}}}$

Step 10.2.4

Cancel the common factor of $2$ .

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Step 10.2.4.1

Cancel the common factor.

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9 \cdot 11^{\frac{2}{2}}}{11^{2}}}$

Step 10.2.4.2

Rewrite the expression.

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9 \cdot 11^{1}}{11^{2}}}$

Step 10.2.5

Evaluate the exponent.

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9 \cdot 11}{11^{2}}}$

Step 11

Reduce the expression by cancelling the common factors.

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Step 11.1

Raise $11$ to the power of $2$ .

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9 \cdot 11}{121}}$

Step 11.2

Multiply $9$ by $11$ .

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{99}{121}}$

Step 11.3

Cancel the common factor of $99$ and $121$ .

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Step 11.3.1

Factor $11$ out of $99$ .

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{11 (9)}{121}}$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Factor $11$ out of $121$ .

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{11 \cdot 9}{11 \cdot 11}}$

Step 11.3.2.2

Cancel the common factor.

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{11 \cdot 9}{11 \cdot 11}}$

Step 11.3.2.3

Rewrite the expression.

$\sqrt{\frac{1}{11} + \frac{1}{11} + \frac{9}{11}}$

Step 11.4

Simplify the expression.

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Step 11.4.1

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 1}{11} + \frac{9}{11}}$

Step 11.4.2

Add $1$ and $1$ .

$\sqrt{\frac{2}{11} + \frac{9}{11}}$

Step 11.4.3

Combine the numerators over the common denominator.

$\sqrt{\frac{2 + 9}{11}}$

Step 11.4.4

Add $2$ and $9$ .

$\sqrt{\frac{11}{11}}$

Step 11.4.5

Divide $11$ by $11$ .

$\sqrt{1}$

Step 11.4.6

Any root of $1$ is $1$ .

$1$