Algebra Examples

$\frac{- 9 b^{- 5}}{{(- 3 b^{- 1} \cdot b^{- 4})}^{3}}$

Step 1

Move $b^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 9}{{(- 3 b^{- 1} \cdot b^{- 4})}^{3} b^{5}}$

Step 2

Multiply $b^{- 1}$ by $b^{- 4}$ by adding the exponents.

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Step 2.1

Move $b^{- 4}$ .

$\frac{- 9}{{(- 3 (b^{- 4} b^{- 1}))}^{3} b^{5}}$

Step 2.2

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

$\frac{- 9}{{(- 3 b^{- 4 - 1})}^{3} b^{5}}$

Step 2.3

Subtract $1$ from $- 4$ .

$\frac{- 9}{{(- 3 b^{- 5})}^{3} b^{5}}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{- 9}{{(- 3)}^{3} {(b^{- 5})}^{3} b^{5}}$

Step 3.2

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

$\frac{- 9}{- 27 {(b^{- 5})}^{3} b^{5}}$

Step 3.3

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

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Step 3.3.1

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

$\frac{- 9}{- 27 b^{- 5 \cdot 3} b^{5}}$

Step 3.3.2

Multiply $- 5$ by $3$ .

$\frac{- 9}{- 27 b^{- 15} b^{5}}$

Step 3.4

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

$\frac{- 9}{- 27 \frac{1}{b^{15}} b^{5}}$

Step 3.5

Combine exponents.

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Step 3.5.1

Combine $- 27$ and $\frac{1}{b^{15}}$ .

$\frac{- 9}{\frac{- 27}{b^{15}} b^{5}}$

Step 3.5.2

Combine $\frac{- 27}{b^{15}}$ and $b^{5}$ .

$\frac{- 9}{\frac{- 27 b^{5}}{b^{15}}}$

Step 3.6

Reduce the expression $\frac{- 27 b^{5}}{b^{15}}$ by cancelling the common factors.

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Step 3.6.1

Factor $b^{5}$ out of $- 27 b^{5}$ .

$\frac{- 9}{\frac{b^{5} \cdot - 27}{b^{15}}}$

Step 3.6.2

Factor $b^{5}$ out of $b^{15}$ .

$\frac{- 9}{\frac{b^{5} \cdot - 27}{b^{5} b^{10}}}$

Step 3.6.3

Cancel the common factor.

$\frac{- 9}{\frac{b^{5} \cdot - 27}{b^{5} b^{10}}}$

Step 3.6.4

Rewrite the expression.

$\frac{- 9}{\frac{- 27}{b^{10}}}$

Step 3.7

Move the negative in front of the fraction.

$\frac{- 9}{- \frac{27}{b^{10}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$- 9 (- \frac{b^{10}}{27})$

Step 5

Cancel the common factor of $9$ .

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Step 5.1

Move the leading negative in $- \frac{b^{10}}{27}$ into the numerator.

$- 9 \frac{- b^{10}}{27}$

Step 5.2

Factor $9$ out of $- 9$ .

$9 (- 1) \frac{- b^{10}}{27}$

Step 5.3

Factor $9$ out of $27$ .

$9 \cdot - 1 \frac{- b^{10}}{9 \cdot 3}$

Step 5.4

Cancel the common factor.

$9 \cdot - 1 \frac{- b^{10}}{9 \cdot 3}$

Step 5.5

Rewrite the expression.

$- \frac{- b^{10}}{3}$

Step 6

Move the negative in front of the fraction.

$- - \frac{b^{10}}{3}$

Step 7

Multiply $- - \frac{b^{10}}{3}$ .

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Step 7.1

Multiply $- 1$ by $- 1$ .

$1 \frac{b^{10}}{3}$

Step 7.2

Multiply $\frac{b^{10}}{3}$ by $1$ .

$\frac{b^{10}}{3}$