Algebra Examples

$42 \sqrt{6} = x \sqrt{14}$

Step 1

Rewrite the equation as $x \sqrt{14} = 42 \sqrt{6}$ .

$x \sqrt{14} = 42 \sqrt{6}$

Step 2

Divide each term in $x \sqrt{14} = 42 \sqrt{6}$ by $\sqrt{14}$ and simplify.

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Step 2.1

Divide each term in $x \sqrt{14} = 42 \sqrt{6}$ by $\sqrt{14}$ .

$\frac{x \sqrt{14}}{\sqrt{14}} = \frac{42 \sqrt{6}}{\sqrt{14}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $\sqrt{14}$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{x \sqrt{14}}{\sqrt{14}} = \frac{42 \sqrt{6}}{\sqrt{14}}$

Step 2.2.1.2

Divide $x$ by $1$ .

$x = \frac{42 \sqrt{6}}{\sqrt{14}}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Combine $\sqrt{6}$ and $\sqrt{14}$ into a single radical.

$x = 42 \sqrt{\frac{6}{14}}$

Step 2.3.2

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

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Step 2.3.2.1

Factor $2$ out of $6$ .

$x = 42 \sqrt{\frac{2 (3)}{14}}$

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

Factor $2$ out of $14$ .

$x = 42 \sqrt{\frac{2 \cdot 3}{2 \cdot 7}}$

Step 2.3.2.2.2

Cancel the common factor.

$x = 42 \sqrt{\frac{2 \cdot 3}{2 \cdot 7}}$

Step 2.3.2.2.3

Rewrite the expression.

$x = 42 \sqrt{\frac{3}{7}}$

Step 2.3.3

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

$x = 42 \frac{\sqrt{3}}{\sqrt{7}}$

Step 2.3.4

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

$x = 42 (\frac{\sqrt{3}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Step 2.3.5

Combine and simplify the denominator.

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Step 2.3.5.1

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

$x = 42 \frac{\sqrt{3} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 2.3.5.2

Raise $\sqrt{7}$ to the power of $1$ .

$x = 42 \frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 2.3.5.3

Raise $\sqrt{7}$ to the power of $1$ .

$x = 42 \frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 2.3.5.4

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

$x = 42 \frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 2.3.5.5

Add $1$ and $1$ .

$x = 42 \frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 2.3.5.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.3.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = 42 \frac{\sqrt{3} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 2.3.5.6.2

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

$x = 42 \frac{\sqrt{3} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 2.3.5.6.3

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

$x = 42 \frac{\sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 2.3.5.6.4

Cancel the common factor of $2$ .

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Step 2.3.5.6.4.1

Cancel the common factor.

$x = 42 \frac{\sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 2.3.5.6.4.2

Rewrite the expression.

$x = 42 \frac{\sqrt{3} \sqrt{7}}{7^{1}}$

Step 2.3.5.6.5

Evaluate the exponent.

$x = 42 \frac{\sqrt{3} \sqrt{7}}{7}$

Step 2.3.6

Simplify the numerator.

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Step 2.3.6.1

Combine using the product rule for radicals.

$x = 42 \frac{\sqrt{3 \cdot 7}}{7}$

Step 2.3.6.2

Multiply $3$ by $7$ .

$x = 42 \frac{\sqrt{21}}{7}$

Step 2.3.7

Cancel the common factor of $7$ .

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Step 2.3.7.1

Factor $7$ out of $42$ .

$x = 7 (6) \frac{\sqrt{21}}{7}$

Step 2.3.7.2

Cancel the common factor.

$x = 7 \cdot 6 \frac{\sqrt{21}}{7}$

Step 2.3.7.3

Rewrite the expression.

$x = 6 \sqrt{21}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 6 \sqrt{21}$

Decimal Form:

$x = 27.49545416 \dots$