Calculus Examples

${(27 a^{- 3} b^{6} c^{3})}^{\frac{1}{3}}$

Step 1

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

${(27 \frac{1}{a^{3}} b^{6} c^{3})}^{\frac{1}{3}}$

Step 2

Combine $27$ and $\frac{1}{a^{3}}$ .

${(\frac{27}{a^{3}} b^{6} c^{3})}^{\frac{1}{3}}$

Step 3

Combine $\frac{27}{a^{3}}$ and $b^{6}$ .

${(\frac{27 b^{6}}{a^{3}} c^{3})}^{\frac{1}{3}}$

Step 4

Combine $\frac{27 b^{6}}{a^{3}}$ and $c^{3}$ .

${(\frac{27 b^{6} c^{3}}{a^{3}})}^{\frac{1}{3}}$

Step 5

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

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Step 5.1

Apply the product rule to $\frac{27 b^{6} c^{3}}{a^{3}}$ .

$\frac{{(27 b^{6} c^{3})}^{\frac{1}{3}}}{{(a^{3})}^{\frac{1}{3}}}$

Step 5.2

Apply the product rule to $27 b^{6} c^{3}$ .

$\frac{{(27 b^{6})}^{\frac{1}{3}} {(c^{3})}^{\frac{1}{3}}}{{(a^{3})}^{\frac{1}{3}}}$

Step 5.3

Apply the product rule to $27 b^{6}$ .

$\frac{27^{\frac{1}{3}} {(b^{6})}^{\frac{1}{3}} {(c^{3})}^{\frac{1}{3}}}{{(a^{3})}^{\frac{1}{3}}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite $27$ as $3^{3}$ .

$\frac{{(3^{3})}^{\frac{1}{3}} {(b^{6})}^{\frac{1}{3}} {(c^{3})}^{\frac{1}{3}}}{{(a^{3})}^{\frac{1}{3}}}$

Step 6.2

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

$\frac{3^{3 (\frac{1}{3})} {(b^{6})}^{\frac{1}{3}} {(c^{3})}^{\frac{1}{3}}}{{(a^{3})}^{\frac{1}{3}}}$

Step 6.3

Cancel the common factor of $3$ .

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Step 6.3.1

Cancel the common factor.

$\frac{3^{3 (\frac{1}{3})} {(b^{6})}^{\frac{1}{3}} {(c^{3})}^{\frac{1}{3}}}{{(a^{3})}^{\frac{1}{3}}}$

Step 6.3.2

Rewrite the expression.

$\frac{3^{1} {(b^{6})}^{\frac{1}{3}} {(c^{3})}^{\frac{1}{3}}}{{(a^{3})}^{\frac{1}{3}}}$

Step 6.4

Evaluate the exponent.

$\frac{3 {(b^{6})}^{\frac{1}{3}} {(c^{3})}^{\frac{1}{3}}}{{(a^{3})}^{\frac{1}{3}}}$

Step 6.5

Multiply the exponents in ${(b^{6})}^{\frac{1}{3}}$ .

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Step 6.5.1

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

$\frac{3 b^{6 (\frac{1}{3})} {(c^{3})}^{\frac{1}{3}}}{{(a^{3})}^{\frac{1}{3}}}$

Step 6.5.2

Cancel the common factor of $3$ .

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Step 6.5.2.1

Factor $3$ out of $6$ .

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

Step 6.5.2.2

Cancel the common factor.

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

Step 6.5.2.3

Rewrite the expression.

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

Step 6.6

Multiply the exponents in ${(c^{3})}^{\frac{1}{3}}$ .

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Step 6.6.1

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

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

Step 6.6.2

Cancel the common factor of $3$ .

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Step 6.6.2.1

Cancel the common factor.

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

Step 6.6.2.2

Rewrite the expression.

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

Step 6.7

Simplify.

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

Step 7

Simplify the denominator.

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Step 7.1

Multiply the exponents in ${(a^{3})}^{\frac{1}{3}}$ .

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Step 7.1.1

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

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

Step 7.1.2

Cancel the common factor of $3$ .

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Step 7.1.2.1

Cancel the common factor.

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

Step 7.1.2.2

Rewrite the expression.

$\frac{3 b^{2} c}{a^{1}}$

Step 7.2

Simplify.

$\frac{3 b^{2} c}{a}$