Algebra Examples

$\frac{7 \sqrt{6}}{\sqrt{14}}$

Step 1

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

$7 \sqrt{\frac{6}{14}}$

Step 2

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

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Step 2.1

Factor $2$ out of $6$ .

$7 \sqrt{\frac{2 (3)}{14}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $2$ out of $14$ .

$7 \sqrt{\frac{2 \cdot 3}{2 \cdot 7}}$

Step 2.2.2

Cancel the common factor.

$7 \sqrt{\frac{2 \cdot 3}{2 \cdot 7}}$

Step 2.2.3

Rewrite the expression.

$7 \sqrt{\frac{3}{7}}$

Step 3

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

$7 \frac{\sqrt{3}}{\sqrt{7}}$

Step 4

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

$7 (\frac{\sqrt{3}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Step 5

Combine and simplify the denominator.

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Step 5.1

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

$7 \frac{\sqrt{3} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 5.2

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

$7 \frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 5.3

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

$7 \frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 5.4

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

$7 \frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 5.5

Add $1$ and $1$ .

$7 \frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

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

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

Step 5.6.3

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

$7 \frac{\sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

$7 \frac{\sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 5.6.4.2

Rewrite the expression.

$7 \frac{\sqrt{3} \sqrt{7}}{7^{1}}$

Step 5.6.5

Evaluate the exponent.

$7 \frac{\sqrt{3} \sqrt{7}}{7}$

Step 6

Simplify the numerator.

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Step 6.1

Combine using the product rule for radicals.

$7 \frac{\sqrt{3 \cdot 7}}{7}$

Step 6.2

Multiply $3$ by $7$ .

$7 \frac{\sqrt{21}}{7}$

Step 7

Cancel the common factor of $7$ .

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Step 7.1

Cancel the common factor.

$7 \frac{\sqrt{21}}{7}$

Step 7.2

Rewrite the expression.

$\sqrt{21}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\sqrt{21}$

Decimal Form:

$4.58257569 \dots$