Basic Math Examples

$8 \sqrt{\frac{5}{32}}$

Step 1

Rewrite $\sqrt{\frac{5}{32}}$ as $\frac{\sqrt{5}}{\sqrt{32}}$ .

$8 \frac{\sqrt{5}}{\sqrt{32}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 2.1.1

Factor $16$ out of $32$ .

$8 \frac{\sqrt{5}}{\sqrt{16 (2)}}$

Step 2.1.2

Rewrite $16$ as $4^{2}$ .

$8 \frac{\sqrt{5}}{\sqrt{4^{2} \cdot 2}}$

Step 2.2

Pull terms out from under the radical.

$8 \frac{\sqrt{5}}{4 \sqrt{2}}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $4$ .

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Step 3.1.1

Factor $4$ out of $8$ .

$4 (2) \frac{\sqrt{5}}{4 \sqrt{2}}$

Step 3.1.2

Factor $4$ out of $4 \sqrt{2}$ .

$4 (2) \frac{\sqrt{5}}{4 (\sqrt{2})}$

Step 3.1.3

Cancel the common factor.

$4 \cdot 2 \frac{\sqrt{5}}{4 \sqrt{2}}$

Step 3.1.4

Rewrite the expression.

$2 \frac{\sqrt{5}}{\sqrt{2}}$

Step 3.2

Combine $2$ and $\frac{\sqrt{5}}{\sqrt{2}}$ .

$\frac{2 \sqrt{5}}{\sqrt{2}}$

Step 4

Multiply $\frac{2 \sqrt{5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{2 \sqrt{5}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{2 \sqrt{5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{2 \sqrt{5} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 \sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 5.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 \sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 5.4

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

$\frac{2 \sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 5.5

Add $1$ and $1$ .

$\frac{2 \sqrt{5} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 5.6

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

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Step 5.6.1

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

$\frac{2 \sqrt{5} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 5.6.2

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

$\frac{2 \sqrt{5} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 5.6.3

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

$\frac{2 \sqrt{5} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{5} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{5} \sqrt{2}}{2^{1}}$

Step 5.6.5

Evaluate the exponent.

$\frac{2 \sqrt{5} \sqrt{2}}{2}$

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Cancel the common factor.

$\frac{2 \sqrt{5} \sqrt{2}}{2}$

Step 6.2

Divide $\sqrt{5} \sqrt{2}$ by $1$ .

$\sqrt{5} \sqrt{2}$

Step 7

Combine using the product rule for radicals.

$\sqrt{5 \cdot 2}$

Step 8

Multiply $5$ by $2$ .

$\sqrt{10}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\sqrt{10}$

Decimal Form:

$3.16227766 \dots$