Precalculus Examples

${(a^{2} b^{8} c^{5})}^{2} {(a^{5} c^{2})}^{3} {(a^{- 2} b c^{- 1})}^{3}$

Step 1

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

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Step 1.1

Apply the product rule to $a^{2} b^{8} c^{5}$ .

${(a^{2} b^{8})}^{2} {(c^{5})}^{2} {(a^{5} c^{2})}^{3} {(a^{- 2} b c^{- 1})}^{3}$

Step 1.2

Apply the product rule to $a^{2} b^{8}$ .

${(a^{2})}^{2} {(b^{8})}^{2} {(c^{5})}^{2} {(a^{5} c^{2})}^{3} {(a^{- 2} b c^{- 1})}^{3}$

Step 2

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

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Step 2.1

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

$a^{2 \cdot 2} {(b^{8})}^{2} {(c^{5})}^{2} {(a^{5} c^{2})}^{3} {(a^{- 2} b c^{- 1})}^{3}$

Step 2.2

Multiply $2$ by $2$ .

$a^{4} {(b^{8})}^{2} {(c^{5})}^{2} {(a^{5} c^{2})}^{3} {(a^{- 2} b c^{- 1})}^{3}$

Step 3

Multiply the exponents in ${(b^{8})}^{2}$ .

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Step 3.1

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

$a^{4} b^{8 \cdot 2} {(c^{5})}^{2} {(a^{5} c^{2})}^{3} {(a^{- 2} b c^{- 1})}^{3}$

Step 3.2

Multiply $8$ by $2$ .

$a^{4} b^{16} {(c^{5})}^{2} {(a^{5} c^{2})}^{3} {(a^{- 2} b c^{- 1})}^{3}$

Step 4

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

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Step 4.1

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

$a^{4} b^{16} c^{5 \cdot 2} {(a^{5} c^{2})}^{3} {(a^{- 2} b c^{- 1})}^{3}$

Step 4.2

Multiply $5$ by $2$ .

$a^{4} b^{16} c^{10} {(a^{5} c^{2})}^{3} {(a^{- 2} b c^{- 1})}^{3}$

Step 5

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

$a^{4} b^{16} c^{10} ({(a^{5})}^{3} {(c^{2})}^{3}) {(a^{- 2} b c^{- 1})}^{3}$

Step 6

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

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Step 6.1

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

$a^{4} b^{16} c^{10} (a^{5 \cdot 3} {(c^{2})}^{3}) {(a^{- 2} b c^{- 1})}^{3}$

Step 6.2

Multiply $5$ by $3$ .

$a^{4} b^{16} c^{10} (a^{15} {(c^{2})}^{3}) {(a^{- 2} b c^{- 1})}^{3}$

Step 7

Multiply $a^{4}$ by $a^{15}$ by adding the exponents.

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Step 7.1

Move $a^{15}$ .

$a^{15} a^{4} b^{16} c^{10} {(c^{2})}^{3} {(a^{- 2} b c^{- 1})}^{3}$

Step 7.2

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

$a^{15 + 4} b^{16} c^{10} {(c^{2})}^{3} {(a^{- 2} b c^{- 1})}^{3}$

Step 7.3

Add $15$ and $4$ .

$a^{19} b^{16} c^{10} {(c^{2})}^{3} {(a^{- 2} b c^{- 1})}^{3}$

Step 8

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

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Step 8.1

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

$a^{19} b^{16} c^{10} c^{2 \cdot 3} {(a^{- 2} b c^{- 1})}^{3}$

Step 8.2

Multiply $2$ by $3$ .

$a^{19} b^{16} c^{10} c^{6} {(a^{- 2} b c^{- 1})}^{3}$

Step 9

Multiply $c^{10}$ by $c^{6}$ by adding the exponents.

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Step 9.1

Move $c^{6}$ .

$a^{19} b^{16} (c^{6} c^{10}) {(a^{- 2} b c^{- 1})}^{3}$

Step 9.2

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

$a^{19} b^{16} c^{6 + 10} {(a^{- 2} b c^{- 1})}^{3}$

Step 9.3

Add $6$ and $10$ .

$a^{19} b^{16} c^{16} {(a^{- 2} b c^{- 1})}^{3}$

Step 10

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

$a^{19} b^{16} c^{16} {(\frac{1}{a^{2}} b c^{- 1})}^{3}$

Step 11

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

$a^{19} b^{16} c^{16} {(\frac{b}{a^{2}} c^{- 1})}^{3}$

Step 12

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

$a^{19} b^{16} c^{16} {(\frac{b}{a^{2}} \cdot \frac{1}{c})}^{3}$

Step 13

Combine.

$a^{19} b^{16} c^{16} {(\frac{b \cdot 1}{a^{2} c})}^{3}$

Step 14

Multiply $b$ by $1$ .

$a^{19} b^{16} c^{16} {(\frac{b}{a^{2} c})}^{3}$

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

$a^{19} b^{16} c^{16} \frac{b^{3}}{{(a^{2} c)}^{3}}$

Step 15.2

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

$a^{19} b^{16} c^{16} \frac{b^{3}}{{(a^{2})}^{3} c^{3}}$

Step 16

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

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Step 16.1

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

$a^{19} b^{16} c^{16} \frac{b^{3}}{a^{2 \cdot 3} c^{3}}$

Step 16.2

Multiply $2$ by $3$ .

$a^{19} b^{16} c^{16} \frac{b^{3}}{a^{6} c^{3}}$

Step 17

Cancel the common factor of $a^{6} c^{3}$ .

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Step 17.1

Factor $a^{6} c^{3}$ out of $a^{19} b^{16} c^{16}$ .

$a^{6} c^{3} (a^{13} b^{16} c^{13}) \frac{b^{3}}{a^{6} c^{3}}$

Step 17.2

Cancel the common factor.

$a^{6} c^{3} (a^{13} b^{16} c^{13}) \frac{b^{3}}{a^{6} c^{3}}$

Step 17.3

Rewrite the expression.

$a^{13} b^{16} c^{13} b^{3}$

Step 18

Multiply $b^{16}$ by $b^{3}$ by adding the exponents.

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Step 18.1

Move $b^{3}$ .

$a^{13} (b^{3} b^{16}) c^{13}$

Step 18.2

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

$a^{13} b^{3 + 16} c^{13}$

Step 18.3

Add $3$ and $16$ .

$a^{13} b^{19} c^{13}$