Precalculus Examples

$\frac{a^{- 1}}{(b^{- 1}) \sqrt{a}}$

Step 1

Move $a^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{(b^{- 1}) \sqrt{a} a}$

Step 2

Move $b^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{b}{\sqrt{a} a}$

Step 3

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

$\frac{b}{\sqrt{a} a} \cdot \frac{\sqrt{a}}{\sqrt{a}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

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

$\frac{b \sqrt{a}}{\sqrt{a} a \sqrt{a}}$

Step 4.2

Move $\sqrt{a}$ .

$\frac{b \sqrt{a}}{a (\sqrt{a} \sqrt{a})}$

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

Add $1$ and $1$ .

$\frac{b \sqrt{a}}{a {\sqrt{a}}^{2}}$

Step 4.7

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

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Step 4.7.1

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

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

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.7.4

Cancel the common factor of $2$ .

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Step 4.7.4.1

Cancel the common factor.

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

Step 4.7.4.2

Rewrite the expression.

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

Step 4.7.5

Simplify.

$\frac{b \sqrt{a}}{a \cdot a}$

Step 5

Multiply $a$ by $a$ .

$\frac{b \sqrt{a}}{a^{2}}$