Simplify the Radical Expression square root of 25/128

$\sqrt{\frac{25}{128}}$

Step 1

Rewrite $\sqrt{\frac{25}{128}}$ as $\frac{\sqrt{25}}{\sqrt{128}}$ .

$\frac{\sqrt{25}}{\sqrt{128}}$

Step 2

Simplify the numerator.

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2}}}{\sqrt{128}}$

Pull terms out from under the radical.

$\frac{| 5 |}{\sqrt{128}}$

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{5}{\sqrt{128}}$

Step 3

Simplify the denominator.

Rewrite $128$ as $8^{2} \cdot 2$ .

Factor $64$ out of $128$ .

$\frac{5}{\sqrt{64 (2)}}$

Rewrite $64$ as $8^{2}$ .

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

Pull terms out from under the radical.

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

The absolute value is the distance between a number and zero. The distance between $0$ and $8$ is $8$ .

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

Step 4

Multiply $\frac{5}{8 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 5

Combine and simplify the denominator.

Multiply $\frac{5}{8 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Move $\sqrt{2}$ .

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

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

$\frac{5 \sqrt{2}}{8 ({\sqrt{2}}^{1} \sqrt{2})}$

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

$\frac{5 \sqrt{2}}{8 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

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

$\frac{5 \sqrt{2}}{8 {\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

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

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

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

$\frac{5 \sqrt{2}}{8 {(2^{\frac{1}{2}})}^{2}}$

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

$\frac{5 \sqrt{2}}{8 \cdot 2^{\frac{1}{2} \cdot 2}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{5 \sqrt{2}}{8 \cdot 2^{1}}$

Evaluate the exponent.

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

Step 6

Multiply $8$ by $2$ .

$\frac{5 \sqrt{2}}{16}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{5 \sqrt{2}}{16}$

Decimal Form:

$0.44194173 \dots$

Simplify the Radical Expression square root of 25/128

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