Precalculus Examples

${(3 a^{- 1} b^{2})}^{3} {(\frac{a^{2}}{b})}^{\frac{1}{2}}$

Step 1

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

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

Step 2

Combine $3$ and $\frac{1}{a}$ .

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

Step 3

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

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

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1

Apply the product rule to $\frac{3 b^{2}}{a}$ .

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

Step 4.2

Apply the product rule to $3 b^{2}$ .

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

Step 5

Simplify the numerator.

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Step 5.1

Raise $3$ to the power of $3$ .

$\frac{27 {(b^{2})}^{3}}{a^{3}} {(\frac{a^{2}}{b})}^{\frac{1}{2}}$

Step 5.2

Multiply the exponents in ${(b^{2})}^{3}$ .

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Step 5.2.1

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

$\frac{27 b^{2 \cdot 3}}{a^{3}} {(\frac{a^{2}}{b})}^{\frac{1}{2}}$

Step 5.2.2

Multiply $2$ by $3$ .

$\frac{27 b^{6}}{a^{3}} {(\frac{a^{2}}{b})}^{\frac{1}{2}}$

Step 6

Combine fractions.

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Step 6.1

Apply the product rule to $\frac{a^{2}}{b}$ .

$\frac{27 b^{6}}{a^{3}} \cdot \frac{{(a^{2})}^{\frac{1}{2}}}{b^{\frac{1}{2}}}$

Step 6.2

Combine.

$\frac{27 b^{6} {(a^{2})}^{\frac{1}{2}}}{a^{3} b^{\frac{1}{2}}}$

Step 6.3

Move $b^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{27 b^{6} {(a^{2})}^{\frac{1}{2}} b^{- \frac{1}{2}}}{a^{3}}$

Step 7

Multiply $b^{6}$ by $b^{- \frac{1}{2}}$ by adding the exponents.

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Step 7.1

Move $b^{- \frac{1}{2}}$ .

$\frac{27 (b^{- \frac{1}{2}} b^{6}) {(a^{2})}^{\frac{1}{2}}}{a^{3}}$

Step 7.2

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

$\frac{27 b^{- \frac{1}{2} + 6} {(a^{2})}^{\frac{1}{2}}}{a^{3}}$

Step 7.3

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{27 b^{- \frac{1}{2} + 6 \cdot \frac{2}{2}} {(a^{2})}^{\frac{1}{2}}}{a^{3}}$

Step 7.4

Combine $6$ and $\frac{2}{2}$ .

$\frac{27 b^{- \frac{1}{2} + \frac{6 \cdot 2}{2}} {(a^{2})}^{\frac{1}{2}}}{a^{3}}$

Step 7.5

Combine the numerators over the common denominator.

$\frac{27 b^{\frac{- 1 + 6 \cdot 2}{2}} {(a^{2})}^{\frac{1}{2}}}{a^{3}}$

Step 7.6

Simplify the numerator.

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Step 7.6.1

Multiply $6$ by $2$ .

$\frac{27 b^{\frac{- 1 + 12}{2}} {(a^{2})}^{\frac{1}{2}}}{a^{3}}$

Step 7.6.2

Add $- 1$ and $12$ .

$\frac{27 b^{\frac{11}{2}} {(a^{2})}^{\frac{1}{2}}}{a^{3}}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply the exponents in ${(a^{2})}^{\frac{1}{2}}$ .

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Step 8.1.1

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

$\frac{27 b^{\frac{11}{2}} a^{2 (\frac{1}{2})}}{a^{3}}$

Step 8.1.2

Cancel the common factor of $2$ .

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Step 8.1.2.1

Cancel the common factor.

$\frac{27 b^{\frac{11}{2}} a^{2 (\frac{1}{2})}}{a^{3}}$

Step 8.1.2.2

Rewrite the expression.

$\frac{27 b^{\frac{11}{2}} a^{1}}{a^{3}}$

Step 8.2

Simplify.

$\frac{27 b^{\frac{11}{2}} a}{a^{3}}$

Step 9

Factor $a$ out of $27 b^{\frac{11}{2}} a$ .

$\frac{a (27 b^{\frac{11}{2}})}{a^{3}}$

Step 10

Cancel the common factors.

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Step 10.1

Factor $a$ out of $a^{3}$ .

$\frac{a (27 b^{\frac{11}{2}})}{a \cdot a^{2}}$

Step 10.2

Cancel the common factor.

$\frac{a (27 b^{\frac{11}{2}})}{a \cdot a^{2}}$

Step 10.3

Rewrite the expression.

$\frac{27 b^{\frac{11}{2}}}{a^{2}}$