Algebra Examples

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

Step 1

Move ${(5 a^{- 4} b^{- 5})}^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2

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

${(5 a^{3} \frac{1}{b^{2}})}^{4} {(5 a^{- 4} b^{- 5})}^{2}$

Step 3

Multiply $5 a^{3} \frac{1}{b^{2}}$ .

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Step 3.1

Combine $5$ and $\frac{1}{b^{2}}$ .

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

Step 3.2

Combine $a^{3}$ and $\frac{5}{b^{2}}$ .

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

Step 4

Move $5$ to the left of $a^{3}$ .

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

Step 5

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

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Step 5.1

Apply the product rule to $\frac{5 a^{3}}{b^{2}}$ .

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

Step 5.2

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

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

Step 6

Simplify the numerator.

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Step 6.1

Raise $5$ to the power of $4$ .

$\frac{625 {(a^{3})}^{4}}{{(b^{2})}^{4}} {(5 a^{- 4} b^{- 5})}^{2}$

Step 6.2

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

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Step 6.2.1

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

$\frac{625 a^{3 \cdot 4}}{{(b^{2})}^{4}} {(5 a^{- 4} b^{- 5})}^{2}$

Step 6.2.2

Multiply $3$ by $4$ .

$\frac{625 a^{12}}{{(b^{2})}^{4}} {(5 a^{- 4} b^{- 5})}^{2}$

Step 7

Multiply the exponents in ${(b^{2})}^{4}$ .

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Step 7.1

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

$\frac{625 a^{12}}{b^{2 \cdot 4}} {(5 a^{- 4} b^{- 5})}^{2}$

Step 7.2

Multiply $2$ by $4$ .

$\frac{625 a^{12}}{b^{8}} {(5 a^{- 4} b^{- 5})}^{2}$

Step 8

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

$\frac{625 a^{12}}{b^{8}} {(5 \frac{1}{a^{4}} b^{- 5})}^{2}$

Step 9

Combine $5$ and $\frac{1}{a^{4}}$ .

$\frac{625 a^{12}}{b^{8}} {(\frac{5}{a^{4}} b^{- 5})}^{2}$

Step 10

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

$\frac{625 a^{12}}{b^{8}} {(\frac{5}{a^{4}} \cdot \frac{1}{b^{5}})}^{2}$

Step 11

Combine.

$\frac{625 a^{12}}{b^{8}} {(\frac{5 \cdot 1}{a^{4} b^{5}})}^{2}$

Step 12

Multiply $5$ by $1$ .

$\frac{625 a^{12}}{b^{8}} {(\frac{5}{a^{4} b^{5}})}^{2}$

Step 13

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

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Step 13.1

Apply the product rule to $\frac{5}{a^{4} b^{5}}$ .

$\frac{625 a^{12}}{b^{8}} \cdot \frac{5^{2}}{{(a^{4} b^{5})}^{2}}$

Step 13.2

Apply the product rule to $a^{4} b^{5}$ .

$\frac{625 a^{12}}{b^{8}} \cdot \frac{5^{2}}{{(a^{4})}^{2} {(b^{5})}^{2}}$

Step 14

Combine.

$\frac{625 a^{12} \cdot 5^{2}}{b^{8} ({(a^{4})}^{2} {(b^{5})}^{2})}$

Step 15

Simplify the numerator.

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Step 15.1

Rewrite $625$ as $5^{4}$ .

$\frac{5^{2} \cdot 5^{4} a^{12}}{b^{8} ({(a^{4})}^{2} {(b^{5})}^{2})}$

Step 15.2

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

$\frac{5^{2 + 4} a^{12}}{b^{8} ({(a^{4})}^{2} {(b^{5})}^{2})}$

Step 15.3

Add $2$ and $4$ .

$\frac{5^{6} a^{12}}{b^{8} ({(a^{4})}^{2} {(b^{5})}^{2})}$

Step 16

Simplify the denominator.

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Step 16.1

Remove unnecessary parentheses.

$\frac{5^{6} a^{12}}{b^{8} {(a^{4})}^{2} {(b^{5})}^{2}}$

Step 16.2

Multiply the exponents in ${(a^{4})}^{2}$ .

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Step 16.2.1

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

$\frac{5^{6} a^{12}}{b^{8} a^{4 \cdot 2} {(b^{5})}^{2}}$

Step 16.2.2

Multiply $4$ by $2$ .

$\frac{5^{6} a^{12}}{b^{8} a^{8} {(b^{5})}^{2}}$

Step 16.3

Multiply the exponents in ${(b^{5})}^{2}$ .

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Step 16.3.1

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

$\frac{5^{6} a^{12}}{b^{8} a^{8} b^{5 \cdot 2}}$

Step 16.3.2

Multiply $5$ by $2$ .

$\frac{5^{6} a^{12}}{b^{8} a^{8} b^{10}}$

Step 16.4

Multiply $b^{8}$ by $b^{10}$ by adding the exponents.

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Step 16.4.1

Move $b^{10}$ .

$\frac{5^{6} a^{12}}{b^{10} b^{8} a^{8}}$

Step 16.4.2

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

$\frac{5^{6} a^{12}}{b^{10 + 8} a^{8}}$

Step 16.4.3

Add $10$ and $8$ .

$\frac{5^{6} a^{12}}{b^{18} a^{8}}$

Step 17

Reduce the expression by cancelling the common factors.

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Step 17.1

Raise $5$ to the power of $6$ .

$\frac{15625 a^{12}}{b^{18} a^{8}}$

Step 17.2

Cancel the common factor of $a^{12}$ and $a^{8}$ .

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Step 17.2.1

Factor $a^{8}$ out of $15625 a^{12}$ .

$\frac{a^{8} (15625 a^{4})}{b^{18} a^{8}}$

Step 17.2.2

Cancel the common factors.

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Step 17.2.2.1

Factor $a^{8}$ out of $b^{18} a^{8}$ .

$\frac{a^{8} (15625 a^{4})}{a^{8} b^{18}}$

Step 17.2.2.2

Cancel the common factor.

$\frac{a^{8} (15625 a^{4})}{a^{8} b^{18}}$

Step 17.2.2.3

Rewrite the expression.

$\frac{15625 a^{4}}{b^{18}}$