Algebra Examples

$\frac{- 3 {(q^{- 2} a^{5})}^{3}}{15 q^{8} a^{5}}$

Step 1

Cancel the common factor of $- 3$ and $15$ .

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Step 1.1

Factor $3$ out of $- 3 {(q^{- 2} a^{5})}^{3}$ .

$\frac{3 (- {(q^{- 2} a^{5})}^{3})}{15 q^{8} a^{5}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $3$ out of $15 q^{8} a^{5}$ .

$\frac{3 (- {(q^{- 2} a^{5})}^{3})}{3 (5 q^{8} a^{5})}$

Step 1.2.2

Cancel the common factor.

$\frac{3 (- {(q^{- 2} a^{5})}^{3})}{3 (5 q^{8} a^{5})}$

Step 1.2.3

Rewrite the expression.

$\frac{- {(q^{- 2} a^{5})}^{3}}{5 q^{8} a^{5}}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the product rule to $q^{- 2} a^{5}$ .

$\frac{- {(q^{- 2})}^{3} {(a^{5})}^{3}}{5 q^{8} a^{5}}$

Step 2.2

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

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Step 2.2.1

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

$\frac{- q^{- 2 \cdot 3} {(a^{5})}^{3}}{5 q^{8} a^{5}}$

Step 2.2.2

Multiply $- 2$ by $3$ .

$\frac{- q^{- 6} {(a^{5})}^{3}}{5 q^{8} a^{5}}$

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

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

$\frac{- \frac{1}{q^{6}} a^{5 \cdot 3}}{5 q^{8} a^{5}}$

Step 2.4.2

Multiply $5$ by $3$ .

$\frac{- \frac{1}{q^{6}} a^{15}}{5 q^{8} a^{5}}$

Step 2.5

Combine $a^{15}$ and $\frac{1}{q^{6}}$ .

$\frac{- \frac{a^{15}}{q^{6}}}{5 q^{8} a^{5}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$- \frac{a^{15}}{q^{6}} \cdot \frac{1}{5 q^{8} a^{5}}$

Step 4

Cancel the common factor of $a^{5}$ .

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Step 4.1

Move the leading negative in $- \frac{a^{15}}{q^{6}}$ into the numerator.

$\frac{- a^{15}}{q^{6}} \cdot \frac{1}{5 q^{8} a^{5}}$

Step 4.2

Factor $a^{5}$ out of $- a^{15}$ .

$\frac{a^{5} (- a^{10})}{q^{6}} \cdot \frac{1}{5 q^{8} a^{5}}$

Step 4.3

Factor $a^{5}$ out of $5 q^{8} a^{5}$ .

$\frac{a^{5} (- a^{10})}{q^{6}} \cdot \frac{1}{a^{5} (5 q^{8})}$

Step 4.4

Cancel the common factor.

$\frac{a^{5} (- a^{10})}{q^{6}} \cdot \frac{1}{a^{5} (5 q^{8})}$

Step 4.5

Rewrite the expression.

$\frac{- a^{10}}{q^{6}} \cdot \frac{1}{5 q^{8}}$

Step 5

Multiply $\frac{- a^{10}}{q^{6}}$ by $\frac{1}{5 q^{8}}$ .

$\frac{- a^{10}}{q^{6} (5 q^{8})}$

Step 6

Multiply $q^{6}$ by $q^{8}$ by adding the exponents.

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Step 6.1

Move $q^{8}$ .

$\frac{- a^{10}}{q^{8} q^{6} \cdot 5}$

Step 6.2

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

$\frac{- a^{10}}{q^{8 + 6} \cdot 5}$

Step 6.3

Add $8$ and $6$ .

$\frac{- a^{10}}{q^{14} \cdot 5}$

Step 7

Move $5$ to the left of $q^{14}$ .

$\frac{- a^{10}}{5 \cdot q^{14}}$

Step 8

Move the negative in front of the fraction.

$- \frac{a^{10}}{5 q^{14}}$