Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine.

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

Step 6

Multiply $b^{3}$ by $1$ .

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Multiply $3$ by $5$ .

$\frac{b^{15}}{{(a^{3})}^{5} {(c^{5})}^{5}} {(a^{5} b^{- 2} c^{- 1})}^{2}$

Step 9

Simplify the denominator.

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Step 9.1

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

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Step 9.1.1

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

$\frac{b^{15}}{a^{3 \cdot 5} {(c^{5})}^{5}} {(a^{5} b^{- 2} c^{- 1})}^{2}$

Step 9.1.2

Multiply $3$ by $5$ .

$\frac{b^{15}}{a^{15} {(c^{5})}^{5}} {(a^{5} b^{- 2} c^{- 1})}^{2}$

Step 9.2

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

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Step 9.2.1

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

$\frac{b^{15}}{a^{15} c^{5 \cdot 5}} {(a^{5} b^{- 2} c^{- 1})}^{2}$

Step 9.2.2

Multiply $5$ by $5$ .

$\frac{b^{15}}{a^{15} c^{25}} {(a^{5} b^{- 2} c^{- 1})}^{2}$

Step 10

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

$\frac{b^{15}}{a^{15} c^{25}} {(a^{5} \frac{1}{b^{2}} c^{- 1})}^{2}$

Step 11

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

$\frac{b^{15}}{a^{15} c^{25}} {(\frac{a^{5}}{b^{2}} c^{- 1})}^{2}$

Step 12

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

$\frac{b^{15}}{a^{15} c^{25}} {(\frac{a^{5}}{b^{2}} \cdot \frac{1}{c})}^{2}$

Step 13

Combine.

$\frac{b^{15}}{a^{15} c^{25}} {(\frac{a^{5} \cdot 1}{b^{2} c})}^{2}$

Step 14

Multiply $a^{5}$ by $1$ .

$\frac{b^{15}}{a^{15} c^{25}} {(\frac{a^{5}}{b^{2} c})}^{2}$

Step 15

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

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Step 15.1

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

$\frac{b^{15}}{a^{15} c^{25}} \cdot \frac{{(a^{5})}^{2}}{{(b^{2} c)}^{2}}$

Step 15.2

Apply the product rule to $b^{2} c$ .

$\frac{b^{15}}{a^{15} c^{25}} \cdot \frac{{(a^{5})}^{2}}{{(b^{2})}^{2} c^{2}}$

Step 16

Combine.

$\frac{b^{15} {(a^{5})}^{2}}{a^{15} c^{25} ({(b^{2})}^{2} c^{2})}$

Step 17

Multiply $c^{25}$ by $c^{2}$ by adding the exponents.

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Step 17.1

Move $c^{2}$ .

$\frac{b^{15} {(a^{5})}^{2}}{a^{15} (c^{2} c^{25}) {(b^{2})}^{2}}$

Step 17.2

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

$\frac{b^{15} {(a^{5})}^{2}}{a^{15} c^{2 + 25} {(b^{2})}^{2}}$

Step 17.3

Add $2$ and $25$ .

$\frac{b^{15} {(a^{5})}^{2}}{a^{15} c^{27} {(b^{2})}^{2}}$

Step 18

Apply basic rules of exponents.

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Step 18.1

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

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Step 18.1.1

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

$\frac{b^{15} a^{5 \cdot 2}}{a^{15} c^{27} {(b^{2})}^{2}}$

Step 18.1.2

Multiply $5$ by $2$ .

$\frac{b^{15} a^{10}}{a^{15} c^{27} {(b^{2})}^{2}}$

Step 18.2

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

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Step 18.2.1

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

$\frac{b^{15} a^{10}}{a^{15} c^{27} b^{2 \cdot 2}}$

Step 18.2.2

Multiply $2$ by $2$ .

$\frac{b^{15} a^{10}}{a^{15} c^{27} b^{4}}$

Step 19

Cancel the common factor of $b^{15}$ and $b^{4}$ .

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Step 19.1

Factor $b^{4}$ out of $b^{15} a^{10}$ .

$\frac{b^{4} (b^{11} a^{10})}{a^{15} c^{27} b^{4}}$

Step 19.2

Cancel the common factors.

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Step 19.2.1

Factor $b^{4}$ out of $a^{15} c^{27} b^{4}$ .

$\frac{b^{4} (b^{11} a^{10})}{b^{4} (a^{15} c^{27})}$

Step 19.2.2

Cancel the common factor.

$\frac{b^{4} (b^{11} a^{10})}{b^{4} (a^{15} c^{27})}$

Step 19.2.3

Rewrite the expression.

$\frac{b^{11} a^{10}}{a^{15} c^{27}}$

Step 20

Cancel the common factor of $a^{10}$ and $a^{15}$ .

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Step 20.1

Factor $a^{10}$ out of $b^{11} a^{10}$ .

$\frac{a^{10} b^{11}}{a^{15} c^{27}}$

Step 20.2

Cancel the common factors.

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Step 20.2.1

Factor $a^{10}$ out of $a^{15} c^{27}$ .

$\frac{a^{10} b^{11}}{a^{10} (a^{5} c^{27})}$

Step 20.2.2

Cancel the common factor.

$\frac{a^{10} b^{11}}{a^{10} (a^{5} c^{27})}$

Step 20.2.3

Rewrite the expression.

$\frac{b^{11}}{a^{5} c^{27}}$