Pre-Algebra Examples

$\frac{12 a^{- 3} b^{5} c}{18 {(a^{2} b^{- 2} c - 1)}^{2}}$

Step 1

Move $a^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{12 b^{5} c}{18 {(a^{2} b^{- 2} c - 1)}^{2} a^{3}}$

Step 2

Cancel the common factor of $12$ and $18$ .

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Step 2.1

Factor $6$ out of $12 b^{5} c$ .

$\frac{6 (2 b^{5} c)}{18 {(a^{2} b^{- 2} c - 1)}^{2} a^{3}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $6$ out of $18 {(a^{2} b^{- 2} c - 1)}^{2} a^{3}$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

Combine $a^{2}$ and $\frac{1}{b^{2}}$ .

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

Step 3.3

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

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

Step 3.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{b^{2}}{b^{2}}$ .

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

Step 3.5

Combine $- 1$ and $\frac{b^{2}}{b^{2}}$ .

$\frac{2 b^{5} c}{3 {(\frac{a^{2} c}{b^{2}} + \frac{- b^{2}}{b^{2}})}^{2} a^{3}}$

Step 3.6

Combine the numerators over the common denominator.

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

Step 3.7

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

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

Step 3.8

Combine exponents.

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Step 3.8.1

Combine $3$ and $\frac{{(a^{2} c - b^{2})}^{2}}{{(b^{2})}^{2}}$ .

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

Step 3.8.2

Combine $\frac{3 {(a^{2} c - b^{2})}^{2}}{{(b^{2})}^{2}}$ and $a^{3}$ .

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

Step 3.9

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

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Step 3.9.1

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

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

Step 3.9.2

Multiply $2$ by $2$ .

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Multiply $2 b^{5} c \frac{b^{4}}{3 {(a^{2} c - b^{2})}^{2} a^{3}}$ .

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Step 5.1

Combine $2$ and $\frac{b^{4}}{3 {(a^{2} c - b^{2})}^{2} a^{3}}$ .

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

Step 5.2

Combine $b^{5}$ and $\frac{2 b^{4}}{3 {(a^{2} c - b^{2})}^{2} a^{3}}$ .

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

Step 5.3

Multiply $b^{5}$ by $b^{4}$ by adding the exponents.

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Step 5.3.1

Move $b^{4}$ .

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

Step 5.3.2

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

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

Step 5.3.3

Add $4$ and $5$ .

$c \frac{b^{9} \cdot 2}{3 {(a^{2} c - b^{2})}^{2} a^{3}}$

Step 5.4

Combine $c$ and $\frac{b^{9} \cdot 2}{3 {(a^{2} c - b^{2})}^{2} a^{3}}$ .

$\frac{c (b^{9} \cdot 2)}{3 {(a^{2} c - b^{2})}^{2} a^{3}}$

Step 6

Move $2$ to the left of $c b^{9}$ .

$\frac{2 \cdot (c b^{9})}{3 {(a^{2} c - b^{2})}^{2} a^{3}}$

Step 7

Reorder factors in $\frac{2 c b^{9}}{3 {(a^{2} c - b^{2})}^{2} a^{3}}$ .

$\frac{2 c b^{9}}{3 a^{3} {(a^{2} c - b^{2})}^{2}}$