Pre-Algebra Examples

${(2 a^{3} b^{4})}^{- 2} {(- a^{- 3} b^{- 5})}^{- 3}$

Step 1

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

$\frac{1}{{(2 a^{3} b^{4})}^{2}} {(- a^{- 3} b^{- 5})}^{- 3}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $2 a^{3} b^{4}$ .

$\frac{1}{{(2 a^{3})}^{2} {(b^{4})}^{2}} {(- a^{- 3} b^{- 5})}^{- 3}$

Step 2.2

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

$\frac{1}{2^{2} {(a^{3})}^{2} {(b^{4})}^{2}} {(- a^{- 3} b^{- 5})}^{- 3}$

Step 2.3

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

$\frac{1}{4 {(a^{3})}^{2} {(b^{4})}^{2}} {(- a^{- 3} b^{- 5})}^{- 3}$

Step 2.4

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

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Step 2.4.1

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

$\frac{1}{4 a^{3 \cdot 2} {(b^{4})}^{2}} {(- a^{- 3} b^{- 5})}^{- 3}$

Step 2.4.2

Multiply $3$ by $2$ .

$\frac{1}{4 a^{6} {(b^{4})}^{2}} {(- a^{- 3} b^{- 5})}^{- 3}$

Step 2.5

Multiply the exponents in ${(b^{4})}^{2}$ .

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Step 2.5.1

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

$\frac{1}{4 a^{6} b^{4 \cdot 2}} {(- a^{- 3} b^{- 5})}^{- 3}$

Step 2.5.2

Multiply $4$ by $2$ .

$\frac{1}{4 a^{6} b^{8}} {(- a^{- 3} b^{- 5})}^{- 3}$

Step 3

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

$\frac{1}{4 a^{6} b^{8}} {(- \frac{1}{a^{3}} b^{- 5})}^{- 3}$

Step 4

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

$\frac{1}{4 a^{6} b^{8}} {(- \frac{1}{a^{3}} \cdot \frac{1}{b^{5}})}^{- 3}$

Step 5

Multiply $\frac{1}{b^{5}}$ by $\frac{1}{a^{3}}$ .

$\frac{1}{4 a^{6} b^{8}} {(- \frac{1}{b^{5} a^{3}})}^{- 3}$

Step 6

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{1}{4 a^{6} b^{8}} {(- b^{5} a^{3})}^{3}$

Step 7

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

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Step 7.1

Apply the product rule to $- b^{5} a^{3}$ .

$\frac{1}{4 a^{6} b^{8}} ({(- b^{5})}^{3} {(a^{3})}^{3})$

Step 7.2

Apply the product rule to $- b^{5}$ .

$\frac{1}{4 a^{6} b^{8}} ({(- 1)}^{3} {(b^{5})}^{3} {(a^{3})}^{3})$

Step 8

Simplify terms.

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Step 8.1

Rewrite using the commutative property of multiplication.

${(- 1)}^{3} \frac{1}{4 a^{6} b^{8}} ({(b^{5})}^{3} {(a^{3})}^{3})$

Step 8.2

Simplify the expression.

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Step 8.2.1

Raise $- 1$ to the power of $3$ .

$- \frac{1}{4 a^{6} b^{8}} ({(b^{5})}^{3} {(a^{3})}^{3})$

Step 8.2.2

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

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Step 8.2.2.1

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

$- \frac{1}{4 a^{6} b^{8}} (b^{5 \cdot 3} {(a^{3})}^{3})$

Step 8.2.2.2

Multiply $5$ by $3$ .

$- \frac{1}{4 a^{6} b^{8}} (b^{15} {(a^{3})}^{3})$

Step 8.3

Cancel the common factor of $b^{8}$ .

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Step 8.3.1

Move the leading negative in $- \frac{1}{4 a^{6} b^{8}}$ into the numerator.

$\frac{- 1}{4 a^{6} b^{8}} (b^{15} {(a^{3})}^{3})$

Step 8.3.2

Factor $b^{8}$ out of $4 a^{6} b^{8}$ .

$\frac{- 1}{b^{8} (4 a^{6})} (b^{15} {(a^{3})}^{3})$

Step 8.3.3

Factor $b^{8}$ out of $b^{15} {(a^{3})}^{3}$ .

$\frac{- 1}{b^{8} (4 a^{6})} (b^{8} (b^{7} {(a^{3})}^{3}))$

Step 8.3.4

Cancel the common factor.

$\frac{- 1}{b^{8} (4 a^{6})} (b^{8} (b^{7} {(a^{3})}^{3}))$

Step 8.3.5

Rewrite the expression.

$\frac{- 1}{4 a^{6}} (b^{7} {(a^{3})}^{3})$

Step 8.4

Combine $b^{7}$ and $\frac{- 1}{4 a^{6}}$ .

$\frac{b^{7} \cdot - 1}{4 a^{6}} {(a^{3})}^{3}$

Step 8.5

Combine $\frac{b^{7} \cdot - 1}{4 a^{6}}$ and ${(a^{3})}^{3}$ .

$\frac{b^{7} \cdot - 1 {(a^{3})}^{3}}{4 a^{6}}$

Step 9

Simplify the numerator.

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Step 9.1

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

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Step 9.1.1

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

$\frac{b^{7} \cdot - 1 a^{3 \cdot 3}}{4 a^{6}}$

Step 9.1.2

Multiply $3$ by $3$ .

$\frac{b^{7} \cdot - 1 a^{9}}{4 a^{6}}$

Step 9.2

Factor out negative.

$\frac{- b^{7} a^{9}}{4 a^{6}}$

Step 10

Reduce the expression by cancelling the common factors.

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Step 10.1

Cancel the common factor of $a^{9}$ and $a^{6}$ .

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Step 10.1.1

Factor $a^{6}$ out of $- b^{7} a^{9}$ .

$\frac{a^{6} (- b^{7} a^{3})}{4 a^{6}}$

Step 10.1.2

Cancel the common factors.

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Step 10.1.2.1

Factor $a^{6}$ out of $4 a^{6}$ .

$\frac{a^{6} (- b^{7} a^{3})}{a^{6} \cdot 4}$

Step 10.1.2.2

Cancel the common factor.

$\frac{a^{6} (- b^{7} a^{3})}{a^{6} \cdot 4}$

Step 10.1.2.3

Rewrite the expression.

$\frac{- b^{7} a^{3}}{4}$

Step 10.2

Move the negative in front of the fraction.

$- \frac{b^{7} a^{3}}{4}$