Algebra Examples

${(\frac{- 3 a^{- 5} c^{4} d^{5}}{a^{5} c^{- 4} d^{9}})}^{- 3}$

Step 1

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

${(\frac{- 3 c^{4} d^{5}}{a^{5} c^{- 4} d^{9} a^{5}})}^{- 3}$

Step 2

Multiply $a^{5}$ by $a^{5}$ by adding the exponents.

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Step 2.1

Move $a^{5}$ .

${(\frac{- 3 c^{4} d^{5}}{a^{5} a^{5} c^{- 4} d^{9}})}^{- 3}$

Step 2.2

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

${(\frac{- 3 c^{4} d^{5}}{a^{5 + 5} c^{- 4} d^{9}})}^{- 3}$

Step 2.3

Add $5$ and $5$ .

${(\frac{- 3 c^{4} d^{5}}{a^{10} c^{- 4} d^{9}})}^{- 3}$

Step 3

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

${(\frac{- 3 c^{4} d^{5} c^{4}}{a^{10} d^{9}})}^{- 3}$

Step 4

Multiply $c^{4}$ by $c^{4}$ by adding the exponents.

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Step 4.1

Move $c^{4}$ .

${(\frac{- 3 (c^{4} c^{4}) d^{5}}{a^{10} d^{9}})}^{- 3}$

Step 4.2

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

${(\frac{- 3 c^{4 + 4} d^{5}}{a^{10} d^{9}})}^{- 3}$

Step 4.3

Add $4$ and $4$ .

${(\frac{- 3 c^{8} d^{5}}{a^{10} d^{9}})}^{- 3}$

Step 5

Cancel the common factor of $d^{5}$ and $d^{9}$ .

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Step 5.1

Factor $d^{5}$ out of $- 3 c^{8} d^{5}$ .

${(\frac{d^{5} (- 3 c^{8})}{a^{10} d^{9}})}^{- 3}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $d^{5}$ out of $a^{10} d^{9}$ .

${(\frac{d^{5} (- 3 c^{8})}{d^{5} (a^{10} d^{4})})}^{- 3}$

Step 5.2.2

Cancel the common factor.

${(\frac{d^{5} (- 3 c^{8})}{d^{5} (a^{10} d^{4})})}^{- 3}$

Step 5.2.3

Rewrite the expression.

${(\frac{- 3 c^{8}}{a^{10} d^{4}})}^{- 3}$

Step 6

Move the negative in front of the fraction.

${(- \frac{3 c^{8}}{a^{10} d^{4}})}^{- 3}$

Step 7

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

${(- \frac{a^{10} d^{4}}{3 c^{8}})}^{3}$

Step 8

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

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Step 8.1

Apply the product rule to $- \frac{a^{10} d^{4}}{3 c^{8}}$ .

${(- 1)}^{3} {(\frac{a^{10} d^{4}}{3 c^{8}})}^{3}$

Step 8.2

Apply the product rule to $\frac{a^{10} d^{4}}{3 c^{8}}$ .

${(- 1)}^{3} \frac{{(a^{10} d^{4})}^{3}}{{(3 c^{8})}^{3}}$

Step 8.3

Apply the product rule to $a^{10} d^{4}$ .

${(- 1)}^{3} \frac{{(a^{10})}^{3} {(d^{4})}^{3}}{{(3 c^{8})}^{3}}$

Step 8.4

Apply the product rule to $3 c^{8}$ .

${(- 1)}^{3} \frac{{(a^{10})}^{3} {(d^{4})}^{3}}{3^{3} {(c^{8})}^{3}}$

Step 9

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

$- \frac{{(a^{10})}^{3} {(d^{4})}^{3}}{3^{3} {(c^{8})}^{3}}$

Step 10

Simplify the numerator.

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Step 10.1

Multiply the exponents in ${(a^{10})}^{3}$ .

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Step 10.1.1

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

$- \frac{a^{10 \cdot 3} {(d^{4})}^{3}}{3^{3} {(c^{8})}^{3}}$

Step 10.1.2

Multiply $10$ by $3$ .

$- \frac{a^{30} {(d^{4})}^{3}}{3^{3} {(c^{8})}^{3}}$

Step 10.2

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

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Step 10.2.1

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

$- \frac{a^{30} d^{4 \cdot 3}}{3^{3} {(c^{8})}^{3}}$

Step 10.2.2

Multiply $4$ by $3$ .

$- \frac{a^{30} d^{12}}{3^{3} {(c^{8})}^{3}}$

Step 11

Simplify the denominator.

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Step 11.1

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

$- \frac{a^{30} d^{12}}{27 {(c^{8})}^{3}}$

Step 11.2

Multiply the exponents in ${(c^{8})}^{3}$ .

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Step 11.2.1

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

$- \frac{a^{30} d^{12}}{27 c^{8 \cdot 3}}$

Step 11.2.2

Multiply $8$ by $3$ .

$- \frac{a^{30} d^{12}}{27 c^{24}}$