Evaluate square root of 10^-7

$\sqrt{10^{- 7}}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\sqrt{\frac{1}{10^{7}}}$

Step 2

Raise $10$ to the power of $7$ .

$\sqrt{\frac{1}{10000000}}$

Step 3

Rewrite $\sqrt{\frac{1}{10000000}}$ as $\frac{\sqrt{1}}{\sqrt{10000000}}$ .

$\frac{\sqrt{1}}{\sqrt{10000000}}$

Step 4

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{10000000}}$

Step 5

Simplify the denominator.

Rewrite $10000000$ as $1000^{2} \cdot 10$ .

Factor $1000000$ out of $10000000$ .

$\frac{1}{\sqrt{1000000 (10)}}$

Rewrite $1000000$ as $1000^{2}$ .

$\frac{1}{\sqrt{1000^{2} \cdot 10}}$

Pull terms out from under the radical.

$\frac{1}{1000 \sqrt{10}}$

Step 6

Multiply $\frac{1}{1000 \sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{1}{1000 \sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 7

Combine and simplify the denominator.

Multiply $\frac{1}{1000 \sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{\sqrt{10}}{1000 \sqrt{10} \sqrt{10}}$

Move $\sqrt{10}$ .

$\frac{\sqrt{10}}{1000 (\sqrt{10} \sqrt{10})}$

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{\sqrt{10}}{1000 ({\sqrt{10}}^{1} \sqrt{10})}$

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{\sqrt{10}}{1000 ({\sqrt{10}}^{1} {\sqrt{10}}^{1})}$

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

$\frac{\sqrt{10}}{1000 {\sqrt{10}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{\sqrt{10}}{1000 {\sqrt{10}}^{2}}$

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{\sqrt{10}}{1000 {(10^{\frac{1}{2}})}^{2}}$

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

$\frac{\sqrt{10}}{1000 \cdot 10^{\frac{1}{2} \cdot 2}}$

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

$\frac{\sqrt{10}}{1000 \cdot 10^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{10}}{1000 \cdot 10^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{\sqrt{10}}{1000 \cdot 10^{1}}$

Evaluate the exponent.

$\frac{\sqrt{10}}{1000 \cdot 10}$

Step 8

Multiply $1000$ by $10$ .

$\frac{\sqrt{10}}{10000}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{10}}{10000}$

Decimal Form:

$0.00031622 \dots$

Evaluate square root of 10^-7

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