Algebra Examples

$\frac{{(a^{3} b^{- 4} c^{0})}^{- 2}}{{(a^{2} b^{5} c^{3})}^{- 4}}$

Step 1

Simplify the expression.

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Step 1.1

Move ${(a^{3} b^{- 4} c^{0})}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(a^{2} b^{5} c^{3})}^{- 4} {(a^{3} b^{- 4} c^{0})}^{2}}$

Step 1.2

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

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Multiply $2$ by $4$ .

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Multiply $5$ by $4$ .

$\frac{a^{8} b^{20} {(c^{3})}^{4}}{{(a^{3} b^{- 4})}^{2}}$

Step 2.5

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

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Step 2.5.1

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

$\frac{a^{8} b^{20} c^{3 \cdot 4}}{{(a^{3} b^{- 4})}^{2}}$

Step 2.5.2

Multiply $3$ by $4$ .

$\frac{a^{8} b^{20} c^{12}}{{(a^{3} b^{- 4})}^{2}}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{a^{8} b^{20} c^{12}}{{(a^{3})}^{2} {(b^{- 4})}^{2}}$

Step 3.2

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

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Step 3.2.1

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

$\frac{a^{8} b^{20} c^{12}}{a^{3 \cdot 2} {(b^{- 4})}^{2}}$

Step 3.2.2

Multiply $3$ by $2$ .

$\frac{a^{8} b^{20} c^{12}}{a^{6} {(b^{- 4})}^{2}}$

Step 3.3

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

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Step 3.3.1

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

$\frac{a^{8} b^{20} c^{12}}{a^{6} b^{- 4 \cdot 2}}$

Step 3.3.2

Multiply $- 4$ by $2$ .

$\frac{a^{8} b^{20} c^{12}}{a^{6} b^{- 8}}$

Step 3.4

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

$\frac{a^{8} b^{20} c^{12}}{a^{6} \frac{1}{b^{8}}}$

Step 4

Combine $a^{6}$ and $\frac{1}{b^{8}}$ .

$\frac{a^{8} b^{20} c^{12}}{\frac{a^{6}}{b^{8}}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$a^{8} b^{20} c^{12} \frac{b^{8}}{a^{6}}$

Step 6

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

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Step 6.1

Factor $a^{6}$ out of $a^{8} b^{20} c^{12}$ .

$a^{6} (a^{2} b^{20} c^{12}) \frac{b^{8}}{a^{6}}$

Step 6.2

Cancel the common factor.

$a^{6} (a^{2} b^{20} c^{12}) \frac{b^{8}}{a^{6}}$

Step 6.3

Rewrite the expression.

$a^{2} b^{20} c^{12} b^{8}$

Step 7

Multiply $b^{20}$ by $b^{8}$ by adding the exponents.

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Step 7.1

Move $b^{8}$ .

$a^{2} (b^{8} b^{20}) c^{12}$

Step 7.2

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

$a^{2} b^{8 + 20} c^{12}$

Step 7.3

Add $8$ and $20$ .

$a^{2} b^{28} c^{12}$