Algebra Examples

$\frac{{(a^{4} b^{3})}^{6}}{{(a^{8} b^{- 2} c^{3})}^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(a^{4})}^{6} {(b^{3})}^{6}}{{(a^{8} b^{- 2} c^{3})}^{3}}$

Step 1.2

Multiply the exponents in ${(a^{4})}^{6}$ .

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Step 1.2.1

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

$\frac{a^{4 \cdot 6} {(b^{3})}^{6}}{{(a^{8} b^{- 2} c^{3})}^{3}}$

Step 1.2.2

Multiply $4$ by $6$ .

$\frac{a^{24} {(b^{3})}^{6}}{{(a^{8} b^{- 2} c^{3})}^{3}}$

Step 1.3

Multiply the exponents in ${(b^{3})}^{6}$ .

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Step 1.3.1

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

$\frac{a^{24} b^{3 \cdot 6}}{{(a^{8} b^{- 2} c^{3})}^{3}}$

Step 1.3.2

Multiply $3$ by $6$ .

$\frac{a^{24} b^{18}}{{(a^{8} b^{- 2} c^{3})}^{3}}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $a^{8} b^{- 2} c^{3}$ .

$\frac{a^{24} b^{18}}{{(a^{8} b^{- 2})}^{3} {(c^{3})}^{3}}$

Step 2.2

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

$\frac{a^{24} b^{18}}{{(a^{8} \frac{1}{b^{2}})}^{3} {(c^{3})}^{3}}$

Step 2.3

Combine $a^{8}$ and $\frac{1}{b^{2}}$ .

$\frac{a^{24} b^{18}}{{(\frac{a^{8}}{b^{2}})}^{3} {(c^{3})}^{3}}$

Step 2.4

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

$\frac{a^{24} b^{18}}{\frac{{(a^{8})}^{3}}{{(b^{2})}^{3}} {(c^{3})}^{3}}$

Step 2.5

Multiply the exponents in ${(a^{8})}^{3}$ .

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Step 2.5.1

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

$\frac{a^{24} b^{18}}{\frac{a^{8 \cdot 3}}{{(b^{2})}^{3}} {(c^{3})}^{3}}$

Step 2.5.2

Multiply $8$ by $3$ .

$\frac{a^{24} b^{18}}{\frac{a^{24}}{{(b^{2})}^{3}} {(c^{3})}^{3}}$

Step 2.6

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

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Step 2.6.1

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

$\frac{a^{24} b^{18}}{\frac{a^{24}}{b^{2 \cdot 3}} {(c^{3})}^{3}}$

Step 2.6.2

Multiply $2$ by $3$ .

$\frac{a^{24} b^{18}}{\frac{a^{24}}{b^{6}} {(c^{3})}^{3}}$

Step 2.7

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

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Step 2.7.1

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

$\frac{a^{24} b^{18}}{\frac{a^{24}}{b^{6}} c^{3 \cdot 3}}$

Step 2.7.2

Multiply $3$ by $3$ .

$\frac{a^{24} b^{18}}{\frac{a^{24}}{b^{6}} c^{9}}$

Step 3

Combine $\frac{a^{24}}{b^{6}}$ and $c^{9}$ .

$\frac{a^{24} b^{18}}{\frac{a^{24} c^{9}}{b^{6}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$a^{24} b^{18} \frac{b^{6}}{a^{24} c^{9}}$

Step 5

Cancel the common factor of $a^{24}$ .

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Step 5.1

Factor $a^{24}$ out of $a^{24} b^{18}$ .

$a^{24} (b^{18}) \frac{b^{6}}{a^{24} c^{9}}$

Step 5.2

Factor $a^{24}$ out of $a^{24} c^{9}$ .

$a^{24} (b^{18}) \frac{b^{6}}{a^{24} (c^{9})}$

Step 5.3

Cancel the common factor.

$a^{24} b^{18} \frac{b^{6}}{a^{24} c^{9}}$

Step 5.4

Rewrite the expression.

$b^{18} \frac{b^{6}}{c^{9}}$

Step 6

Combine $b^{18}$ and $\frac{b^{6}}{c^{9}}$ .

$\frac{b^{18} b^{6}}{c^{9}}$

Step 7

Multiply $b^{18}$ by $b^{6}$ by adding the exponents.

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Step 7.1

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

$\frac{b^{18 + 6}}{c^{9}}$

Step 7.2

Add $18$ and $6$ .

$\frac{b^{24}}{c^{9}}$