Algebra Examples

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

Step 1

Move $b^{- 9}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Move ${(3 b^{- 4} c^{- 3})}^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 3

Simplify the numerator.

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Step 3.1

Apply the product rule to $3 b^{- 4} c^{- 3}$ .

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

Step 3.2

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

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

Step 3.3

Combine $3$ and $\frac{1}{b^{4}}$ .

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

Step 3.4

Apply the product rule to $\frac{3}{b^{4}}$ .

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

Step 3.5

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

$\frac{3 \frac{27}{{(b^{4})}^{3}} {(c^{- 3})}^{3}}{b^{9}}$

Step 3.6

Multiply the exponents in ${(b^{4})}^{3}$ .

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Step 3.6.1

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

$\frac{3 \frac{27}{b^{4 \cdot 3}} {(c^{- 3})}^{3}}{b^{9}}$

Step 3.6.2

Multiply $4$ by $3$ .

$\frac{3 \frac{27}{b^{12}} {(c^{- 3})}^{3}}{b^{9}}$

Step 3.7

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

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Step 3.7.1

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

$\frac{3 \frac{27}{b^{12}} c^{- 3 \cdot 3}}{b^{9}}$

Step 3.7.2

Multiply $- 3$ by $3$ .

$\frac{3 \frac{27}{b^{12}} c^{- 9}}{b^{9}}$

Step 3.8

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

$\frac{3 \frac{27}{b^{12}} \frac{1}{c^{9}}}{b^{9}}$

Step 3.9

Combine exponents.

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Step 3.9.1

Combine $3$ and $\frac{27}{b^{12}}$ .

$\frac{\frac{3 \cdot 27}{b^{12}} \cdot \frac{1}{c^{9}}}{b^{9}}$

Step 3.9.2

Multiply $3$ by $27$ .

$\frac{\frac{81}{b^{12}} \cdot \frac{1}{c^{9}}}{b^{9}}$

Step 3.9.3

Multiply $\frac{81}{b^{12}}$ by $\frac{1}{c^{9}}$ .

$\frac{\frac{81}{b^{12} c^{9}}}{b^{9}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{81}{b^{12} c^{9}} \cdot \frac{1}{b^{9}}$

Step 5

Combine.

$\frac{81 \cdot 1}{b^{12} c^{9} b^{9}}$

Step 6

Multiply $b^{12}$ by $b^{9}$ by adding the exponents.

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Step 6.1

Move $b^{9}$ .

$\frac{81 \cdot 1}{b^{9} b^{12} c^{9}}$

Step 6.2

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

$\frac{81 \cdot 1}{b^{9 + 12} c^{9}}$

Step 6.3

Add $9$ and $12$ .

$\frac{81 \cdot 1}{b^{21} c^{9}}$

Step 7

Multiply $81$ by $1$ .

$\frac{81}{b^{21} c^{9}}$