Pre-Algebra Examples

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

Step 1

Move ${(a^{- 2} b^{- 3} c^{- 4})}^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Move ${(a^{2} b^{3} c^{4})}^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 3

Simplify the numerator.

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Step 3.1

Apply the product rule to $a^{2} b^{3} c^{4}$ .

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Multiply $2$ by $4$ .

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Multiply $3$ by $4$ .

$\frac{a^{8} b^{12} {(c^{4})}^{4}}{{(a^{- 2} b^{- 3} c^{- 4})}^{5}}$

Step 3.5

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

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Step 3.5.1

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

$\frac{a^{8} b^{12} c^{4 \cdot 4}}{{(a^{- 2} b^{- 3} c^{- 4})}^{5}}$

Step 3.5.2

Multiply $4$ by $4$ .

$\frac{a^{8} b^{12} c^{16}}{{(a^{- 2} b^{- 3} c^{- 4})}^{5}}$

Step 4

Simplify the denominator.

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Step 4.1

Apply the product rule to $a^{- 2} b^{- 3} c^{- 4}$ .

$\frac{a^{8} b^{12} c^{16}}{{(a^{- 2} b^{- 3})}^{5} {(c^{- 4})}^{5}}$

Step 4.2

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

$\frac{a^{8} b^{12} c^{16}}{{(\frac{1}{a^{2}} b^{- 3})}^{5} {(c^{- 4})}^{5}}$

Step 4.3

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

$\frac{a^{8} b^{12} c^{16}}{{(\frac{1}{a^{2}} \cdot \frac{1}{b^{3}})}^{5} {(c^{- 4})}^{5}}$

Step 4.4

Combine.

$\frac{a^{8} b^{12} c^{16}}{{(\frac{1 \cdot 1}{a^{2} b^{3}})}^{5} {(c^{- 4})}^{5}}$

Step 4.5

Multiply $1$ by $1$ .

$\frac{a^{8} b^{12} c^{16}}{{(\frac{1}{a^{2} b^{3}})}^{5} {(c^{- 4})}^{5}}$

Step 4.6

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

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Step 4.6.1

Apply the product rule to $\frac{1}{a^{2} b^{3}}$ .

$\frac{a^{8} b^{12} c^{16}}{\frac{1^{5}}{{(a^{2} b^{3})}^{5}} {(c^{- 4})}^{5}}$

Step 4.6.2

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

$\frac{a^{8} b^{12} c^{16}}{\frac{1^{5}}{{(a^{2})}^{5} {(b^{3})}^{5}} {(c^{- 4})}^{5}}$

Step 4.7

One to any power is one.

$\frac{a^{8} b^{12} c^{16}}{\frac{1}{{(a^{2})}^{5} {(b^{3})}^{5}} {(c^{- 4})}^{5}}$

Step 4.8

Simplify the denominator.

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Step 4.8.1

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

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Step 4.8.1.1

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

$\frac{a^{8} b^{12} c^{16}}{\frac{1}{a^{2 \cdot 5} {(b^{3})}^{5}} {(c^{- 4})}^{5}}$

Step 4.8.1.2

Multiply $2$ by $5$ .

$\frac{a^{8} b^{12} c^{16}}{\frac{1}{a^{10} {(b^{3})}^{5}} {(c^{- 4})}^{5}}$

Step 4.8.2

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

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Step 4.8.2.1

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

$\frac{a^{8} b^{12} c^{16}}{\frac{1}{a^{10} b^{3 \cdot 5}} {(c^{- 4})}^{5}}$

Step 4.8.2.2

Multiply $3$ by $5$ .

$\frac{a^{8} b^{12} c^{16}}{\frac{1}{a^{10} b^{15}} {(c^{- 4})}^{5}}$

Step 4.9

Multiply the exponents in ${(c^{- 4})}^{5}$ .

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Step 4.9.1

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

$\frac{a^{8} b^{12} c^{16}}{\frac{1}{a^{10} b^{15}} c^{- 4 \cdot 5}}$

Step 4.9.2

Multiply $- 4$ by $5$ .

$\frac{a^{8} b^{12} c^{16}}{\frac{1}{a^{10} b^{15}} c^{- 20}}$

Step 4.10

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

$\frac{a^{8} b^{12} c^{16}}{\frac{1}{a^{10} b^{15}} \cdot \frac{1}{c^{20}}}$

Step 5

Multiply $\frac{1}{a^{10} b^{15}}$ by $\frac{1}{c^{20}}$ .

$\frac{a^{8} b^{12} c^{16}}{\frac{1}{a^{10} b^{15} c^{20}}}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$a^{8} b^{12} c^{16} (a^{10} b^{15} c^{20})$

Step 7

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

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Step 7.1

Move $a^{10}$ .

$a^{10} a^{8} b^{12} c^{16} (b^{15} c^{20})$

Step 7.2

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

$a^{10 + 8} b^{12} c^{16} (b^{15} c^{20})$

Step 7.3

Add $10$ and $8$ .

$a^{18} b^{12} c^{16} (b^{15} c^{20})$

Step 8

Multiply $b^{12}$ by $b^{15}$ by adding the exponents.

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Step 8.1

Move $b^{15}$ .

$a^{18} (b^{15} b^{12}) c^{16} c^{20}$

Step 8.2

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

$a^{18} b^{15 + 12} c^{16} c^{20}$

Step 8.3

Add $15$ and $12$ .

$a^{18} b^{27} c^{16} c^{20}$

Step 9

Multiply $c^{16}$ by $c^{20}$ by adding the exponents.

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Step 9.1

Move $c^{20}$ .

$a^{18} b^{27} (c^{20} c^{16})$

Step 9.2

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

$a^{18} b^{27} c^{20 + 16}$

Step 9.3

Add $20$ and $16$ .

$a^{18} b^{27} c^{36}$