Precalculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Multiply $2$ by $5$ .

$\frac{a^{10}}{b^{5}} {(\frac{a^{3} b^{2}}{c^{3}})}^{3}$

Step 3

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

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Step 3.1

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

$\frac{a^{10}}{b^{5}} \cdot \frac{{(a^{3} b^{2})}^{3}}{{(c^{3})}^{3}}$

Step 3.2

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

$\frac{a^{10}}{b^{5}} \cdot \frac{{(a^{3})}^{3} {(b^{2})}^{3}}{{(c^{3})}^{3}}$

Step 4

Combine.

$\frac{a^{10} ({(a^{3})}^{3} {(b^{2})}^{3})}{b^{5} {(c^{3})}^{3}}$

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

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

$\frac{a^{10} a^{3 \cdot 3} {(b^{2})}^{3}}{b^{5} {(c^{3})}^{3}}$

Step 5.1.2

Multiply $3$ by $3$ .

$\frac{a^{10} a^{9} {(b^{2})}^{3}}{b^{5} {(c^{3})}^{3}}$

Step 5.2

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

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Step 5.2.1

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

$\frac{a^{10} a^{9} b^{2 \cdot 3}}{b^{5} {(c^{3})}^{3}}$

Step 5.2.2

Multiply $2$ by $3$ .

$\frac{a^{10} a^{9} b^{6}}{b^{5} {(c^{3})}^{3}}$

Step 5.3

Multiply $a^{10}$ by $a^{9}$ by adding the exponents.

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Step 5.3.1

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

$\frac{a^{10 + 9} b^{6}}{b^{5} {(c^{3})}^{3}}$

Step 5.3.2

Add $10$ and $9$ .

$\frac{a^{19} b^{6}}{b^{5} {(c^{3})}^{3}}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

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

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Step 6.1.1

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

$\frac{a^{19} b^{6}}{b^{5} c^{3 \cdot 3}}$

Step 6.1.2

Multiply $3$ by $3$ .

$\frac{a^{19} b^{6}}{b^{5} c^{9}}$

Step 6.2

Cancel the common factor of $b^{6}$ and $b^{5}$ .

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Step 6.2.1

Factor $b^{5}$ out of $a^{19} b^{6}$ .

$\frac{b^{5} (a^{19} b)}{b^{5} c^{9}}$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $b^{5}$ out of $b^{5} c^{9}$ .

$\frac{b^{5} (a^{19} b)}{b^{5} (c^{9})}$

Step 6.2.2.2

Cancel the common factor.

$\frac{b^{5} (a^{19} b)}{b^{5} c^{9}}$

Step 6.2.2.3

Rewrite the expression.

$\frac{a^{19} b}{c^{9}}$