Precalculus Examples

$\sqrt{\frac{12}{7}}$

Step 1

Rewrite $\sqrt{\frac{12}{7}}$ as $\frac{\sqrt{12}}{\sqrt{7}}$ .

$\frac{\sqrt{12}}{\sqrt{7}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.1.1

Factor $4$ out of $12$ .

$\frac{\sqrt{4 (3)}}{\sqrt{7}}$

Step 2.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.2

Pull terms out from under the radical.

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

Step 3

Multiply $\frac{2 \sqrt{3}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

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

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{2 \sqrt{3}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

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

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

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

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

Step 4.6.5

Evaluate the exponent.

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

Step 5

Simplify the numerator.

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Step 5.1

Combine using the product rule for radicals.

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

Step 5.2

Multiply $7$ by $3$ .

$\frac{2 \sqrt{21}}{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{21}}{7}$

Decimal Form:

$1.30930734 \dots$