Precalculus Examples

${(- a b^{3})}^{5} {(- 27 a b^{- 3} b^{9})}^{2}$

Step 1

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

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Step 1.1

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

${(- a)}^{5} {(b^{3})}^{5} {(- 27 a b^{- 3} b^{9})}^{2}$

Step 1.2

Apply the product rule to $- a$ .

${(- 1)}^{5} a^{5} {(b^{3})}^{5} {(- 27 a b^{- 3} b^{9})}^{2}$

Step 2

Raise $- 1$ to the power of $5$ .

$- a^{5} {(b^{3})}^{5} {(- 27 a b^{- 3} b^{9})}^{2}$

Step 3

Multiply the exponents in ${(b^{3})}^{5}$ .

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Step 3.1

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

$- a^{5} b^{3 \cdot 5} {(- 27 a b^{- 3} b^{9})}^{2}$

Step 3.2

Multiply $3$ by $5$ .

$- a^{5} b^{15} {(- 27 a b^{- 3} b^{9})}^{2}$

Step 4

Multiply $b^{- 3}$ by $b^{9}$ by adding the exponents.

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Step 4.1

Move $b^{9}$ .

$- a^{5} b^{15} {(- 27 a (b^{9} b^{- 3}))}^{2}$

Step 4.2

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

$- a^{5} b^{15} {(- 27 a b^{9 - 3})}^{2}$

Step 4.3

Subtract $3$ from $9$ .

$- a^{5} b^{15} {(- 27 a b^{6})}^{2}$

Step 5

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

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Step 5.1

Apply the product rule to $- 27 a b^{6}$ .

$- a^{5} b^{15} ({(- 27 a)}^{2} {(b^{6})}^{2})$

Step 5.2

Apply the product rule to $- 27 a$ .

$- a^{5} b^{15} ({(- 27)}^{2} a^{2} {(b^{6})}^{2})$

Step 6

Multiply $a^{5}$ by $a^{2}$ by adding the exponents.

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Step 6.1

Move $a^{2}$ .

$- (a^{2} a^{5}) b^{15} ({(- 27)}^{2} {(b^{6})}^{2})$

Step 6.2

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

$- a^{2 + 5} b^{15} ({(- 27)}^{2} {(b^{6})}^{2})$

Step 6.3

Add $2$ and $5$ .

$- a^{7} b^{15} ({(- 27)}^{2} {(b^{6})}^{2})$

Step 7

Raise $- 27$ to the power of $2$ .

$- a^{7} b^{15} (729 {(b^{6})}^{2})$

Step 8

Multiply the exponents in ${(b^{6})}^{2}$ .

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Step 8.1

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

$- a^{7} b^{15} (729 b^{6 \cdot 2})$

Step 8.2

Multiply $6$ by $2$ .

$- a^{7} b^{15} (729 b^{12})$

Step 9

Multiply $b^{15}$ by $b^{12}$ by adding the exponents.

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Step 9.1

Move $b^{12}$ .

$- a^{7} (b^{12} b^{15}) \cdot 729$

Step 9.2

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

$- a^{7} b^{12 + 15} \cdot 729$

Step 9.3

Add $12$ and $15$ .

$- a^{7} b^{27} \cdot 729$

Step 10

Multiply $729$ by $- 1$ .

$- 729 a^{7} b^{27}$