Algebra Examples

$\sqrt{a^{- 3}}$

Step 1

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

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

Step 2

Rewrite $\frac{1}{a^{3}}$ as ${(\frac{1}{a})}^{2} \frac{1}{a}$ .

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Step 2.1

Factor the perfect power $1^{2}$ out of $1$ .

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

Step 2.2

Factor the perfect power $a^{2}$ out of $a^{3}$ .

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

Step 2.3

Rearrange the fraction $\frac{1^{2} \cdot 1}{a^{2} a}$ .

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

Step 3

Pull terms out from under the radical.

$| \frac{1}{a} | \sqrt{\frac{1}{a}}$

Step 4

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

$| \frac{1}{a} | \frac{\sqrt{1}}{\sqrt{a}}$

Step 5

Any root of $1$ is $1$ .

$| \frac{1}{a} | \frac{1}{\sqrt{a}}$

Step 6

Multiply $\frac{1}{\sqrt{a}}$ by $\frac{\sqrt{a}}{\sqrt{a}}$ .

$| \frac{1}{a} | (\frac{1}{\sqrt{a}} \cdot \frac{\sqrt{a}}{\sqrt{a}})$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{1}{\sqrt{a}}$ by $\frac{\sqrt{a}}{\sqrt{a}}$ .

$| \frac{1}{a} | \frac{\sqrt{a}}{\sqrt{a} \sqrt{a}}$

Step 7.2

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

$| \frac{1}{a} | \frac{\sqrt{a}}{{\sqrt{a}}^{1} \sqrt{a}}$

Step 7.3

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

$| \frac{1}{a} | \frac{\sqrt{a}}{{\sqrt{a}}^{1} {\sqrt{a}}^{1}}$

Step 7.4

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

$| \frac{1}{a} | \frac{\sqrt{a}}{{\sqrt{a}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

$| \frac{1}{a} | \frac{\sqrt{a}}{{\sqrt{a}}^{2}}$

Step 7.6

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

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Step 7.6.1

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

$| \frac{1}{a} | \frac{\sqrt{a}}{{(a^{\frac{1}{2}})}^{2}}$

Step 7.6.2

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

$| \frac{1}{a} | \frac{\sqrt{a}}{a^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

$| \frac{1}{a} | \frac{\sqrt{a}}{a^{\frac{2}{2}}}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$| \frac{1}{a} | \frac{\sqrt{a}}{a^{\frac{2}{2}}}$

Step 7.6.4.2

Rewrite the expression.

$| \frac{1}{a} | \frac{\sqrt{a}}{a^{1}}$

Step 7.6.5

Simplify.

$| \frac{1}{a} | \frac{\sqrt{a}}{a}$

Step 8

Combine $| \frac{1}{a} |$ and $\frac{\sqrt{a}}{a}$ .

$\frac{| \frac{1}{a} | \sqrt{a}}{a}$