Algebra Examples

$\frac{\sqrt{2}}{\sqrt{7}} = \frac{x}{\sqrt{14}}$

Step 1

Rewrite the equation as $\frac{x}{\sqrt{14}} = \frac{\sqrt{2}}{\sqrt{7}}$ .

$\frac{x}{\sqrt{14}} = \frac{\sqrt{2}}{\sqrt{7}}$

Step 2

Multiply both sides of the equation by $\sqrt{14}$ .

$\sqrt{14} \frac{x}{\sqrt{14}} = \sqrt{14} \frac{\sqrt{2}}{\sqrt{7}}$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

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

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Step 3.1.1.1

Cancel the common factor.

$\sqrt{14} \frac{x}{\sqrt{14}} = \sqrt{14} \frac{\sqrt{2}}{\sqrt{7}}$

Step 3.1.1.2

Rewrite the expression.

$x = \sqrt{14} \frac{\sqrt{2}}{\sqrt{7}}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\sqrt{14} \frac{\sqrt{2}}{\sqrt{7}}$ .

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Step 3.2.1.1

Multiply $\frac{\sqrt{2}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \sqrt{14} (\frac{\sqrt{2}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Step 3.2.1.2

Combine and simplify the denominator.

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Step 3.2.1.2.1

Multiply $\frac{\sqrt{2}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \sqrt{14} \frac{\sqrt{2} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 3.2.1.2.2

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

$x = \sqrt{14} \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 3.2.1.2.3

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

$x = \sqrt{14} \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 3.2.1.2.4

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

$x = \sqrt{14} \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 3.2.1.2.5

Add $1$ and $1$ .

$x = \sqrt{14} \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 3.2.1.2.6

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

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Step 3.2.1.2.6.1

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

$x = \sqrt{14} \frac{\sqrt{2} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 3.2.1.2.6.2

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

$x = \sqrt{14} \frac{\sqrt{2} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 3.2.1.2.6.3

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

$x = \sqrt{14} \frac{\sqrt{2} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 3.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 3.2.1.2.6.4.1

Cancel the common factor.

$x = \sqrt{14} \frac{\sqrt{2} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 3.2.1.2.6.4.2

Rewrite the expression.

$x = \sqrt{14} \frac{\sqrt{2} \sqrt{7}}{7^{1}}$

Step 3.2.1.2.6.5

Evaluate the exponent.

$x = \sqrt{14} \frac{\sqrt{2} \sqrt{7}}{7}$

Step 3.2.1.3

Simplify the numerator.

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Step 3.2.1.3.1

Combine using the product rule for radicals.

$x = \sqrt{14} \frac{\sqrt{2 \cdot 7}}{7}$

Step 3.2.1.3.2

Multiply $2$ by $7$ .

$x = \sqrt{14} \frac{\sqrt{14}}{7}$

Step 3.2.1.4

Multiply $\sqrt{14} \frac{\sqrt{14}}{7}$ .

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Step 3.2.1.4.1

Combine $\sqrt{14}$ and $\frac{\sqrt{14}}{7}$ .

$x = \frac{\sqrt{14} \sqrt{14}}{7}$

Step 3.2.1.4.2

Raise $\sqrt{14}$ to the power of $1$ .

$x = \frac{{\sqrt{14}}^{1} \sqrt{14}}{7}$

Step 3.2.1.4.3

Raise $\sqrt{14}$ to the power of $1$ .

$x = \frac{{\sqrt{14}}^{1} {\sqrt{14}}^{1}}{7}$

Step 3.2.1.4.4

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

$x = \frac{{\sqrt{14}}^{1 + 1}}{7}$

Step 3.2.1.4.5

Add $1$ and $1$ .

$x = \frac{{\sqrt{14}}^{2}}{7}$

Step 3.2.1.5

Rewrite ${\sqrt{14}}^{2}$ as $14$ .

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Step 3.2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$x = \frac{{(14^{\frac{1}{2}})}^{2}}{7}$

Step 3.2.1.5.2

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

$x = \frac{14^{\frac{1}{2} \cdot 2}}{7}$

Step 3.2.1.5.3

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

$x = \frac{14^{\frac{2}{2}}}{7}$

Step 3.2.1.5.4

Cancel the common factor of $2$ .

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Step 3.2.1.5.4.1

Cancel the common factor.

$x = \frac{14^{\frac{2}{2}}}{7}$

Step 3.2.1.5.4.2

Rewrite the expression.

$x = \frac{14^{1}}{7}$

Step 3.2.1.5.5

Evaluate the exponent.

$x = \frac{14}{7}$

Step 3.2.1.6

Divide $14$ by $7$ .

$x = 2$