Linear Algebra Examples

$\sqrt{a^{- 2}} \sqrt[3]{a^{5} b^{- 3}}$

Step 1

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

$\sqrt{\frac{1}{a^{2}}} \sqrt[3]{a^{5} b^{- 3}}$

Step 2

Rewrite $1$ as $1^{2}$ .

$\sqrt{\frac{1^{2}}{a^{2}}} \sqrt[3]{a^{5} b^{- 3}}$

Step 3

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

$\sqrt{{(\frac{1}{a})}^{2}} \sqrt[3]{a^{5} b^{- 3}}$

Step 4

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{a} \sqrt[3]{a^{5} b^{- 3}}$

Step 5

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

$\frac{1}{a} \sqrt[3]{a^{5} \frac{1}{b^{3}}}$

Step 6

Combine $a^{5}$ and $\frac{1}{b^{3}}$ .

$\frac{1}{a} \sqrt[3]{\frac{a^{5}}{b^{3}}}$

Step 7

Rewrite $\frac{a^{5}}{b^{3}}$ as ${(\frac{a}{b})}^{3} a^{2}$ .

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Step 7.1

Factor the perfect power $a^{3}$ out of $a^{5}$ .

$\frac{1}{a} \sqrt[3]{\frac{a^{3} a^{2}}{b^{3}}}$

Step 7.2

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

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

Step 7.3

Rearrange the fraction $\frac{a^{3} a^{2}}{b^{3} \cdot 1}$ .

$\frac{1}{a} \sqrt[3]{{(\frac{a}{b})}^{3} a^{2}}$

Step 8

Pull terms out from under the radical.

$\frac{1}{a} (\frac{a}{b} \sqrt[3]{a^{2}})$

Step 9

Combine $\frac{a}{b}$ and $\sqrt[3]{a^{2}}$ .

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

Step 10

Combine.

$\frac{1 (a \sqrt[3]{a^{2}})}{a b}$

Step 11

Cancel the common factor of $a$ .

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Step 11.1

Cancel the common factor.

$\frac{1 (a \sqrt[3]{a^{2}})}{a b}$

Step 11.2

Rewrite the expression.

$\frac{1 (\sqrt[3]{a^{2}})}{b}$

Step 12

Multiply $\sqrt[3]{a^{2}}$ by $1$ .

$\frac{\sqrt[3]{a^{2}}}{b}$