Calculus Examples

${(\frac{8}{27})}^{- \frac{1}{3}} {(\frac{81}{256})}^{- \frac{1}{4}}$

Step 1

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{27}{8})}^{\frac{1}{3}} {(\frac{81}{256})}^{- \frac{1}{4}}$

Step 2

Apply the product rule to $\frac{27}{8}$ .

$\frac{27^{\frac{1}{3}}}{8^{\frac{1}{3}}} {(\frac{81}{256})}^{- \frac{1}{4}}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{{(3^{3})}^{\frac{1}{3}}}{8^{\frac{1}{3}}} {(\frac{81}{256})}^{- \frac{1}{4}}$

Step 3.2

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

$\frac{3^{3 (\frac{1}{3})}}{8^{\frac{1}{3}}} {(\frac{81}{256})}^{- \frac{1}{4}}$

Step 3.3

Cancel the common factor of $3$ .

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Step 3.3.1

Cancel the common factor.

$\frac{3^{3 (\frac{1}{3})}}{8^{\frac{1}{3}}} {(\frac{81}{256})}^{- \frac{1}{4}}$

Step 3.3.2

Rewrite the expression.

$\frac{3^{1}}{8^{\frac{1}{3}}} {(\frac{81}{256})}^{- \frac{1}{4}}$

Step 3.4

Evaluate the exponent.

$\frac{3}{8^{\frac{1}{3}}} {(\frac{81}{256})}^{- \frac{1}{4}}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $8$ as $2^{3}$ .

$\frac{3}{{(2^{3})}^{\frac{1}{3}}} {(\frac{81}{256})}^{- \frac{1}{4}}$

Step 4.2

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

$\frac{3}{2^{3 (\frac{1}{3})}} {(\frac{81}{256})}^{- \frac{1}{4}}$

Step 4.3

Cancel the common factor of $3$ .

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Step 4.3.1

Cancel the common factor.

$\frac{3}{2^{3 (\frac{1}{3})}} {(\frac{81}{256})}^{- \frac{1}{4}}$

Step 4.3.2

Rewrite the expression.

$\frac{3}{2^{1}} {(\frac{81}{256})}^{- \frac{1}{4}}$

Step 4.4

Evaluate the exponent.

$\frac{3}{2} {(\frac{81}{256})}^{- \frac{1}{4}}$

Step 5

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{3}{2} {(\frac{256}{81})}^{\frac{1}{4}}$

Step 6

Combine fractions.

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Step 6.1

Apply the product rule to $\frac{256}{81}$ .

$\frac{3}{2} \cdot \frac{256^{\frac{1}{4}}}{81^{\frac{1}{4}}}$

Step 6.2

Combine.

$\frac{3 \cdot 256^{\frac{1}{4}}}{2 \cdot 81^{\frac{1}{4}}}$

Step 7

Simplify the numerator.

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Step 7.1

Rewrite $256$ as $4^{4}$ .

$\frac{3 \cdot {(4^{4})}^{\frac{1}{4}}}{2 \cdot 81^{\frac{1}{4}}}$

Step 7.2

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

$\frac{3 \cdot 4^{4 (\frac{1}{4})}}{2 \cdot 81^{\frac{1}{4}}}$

Step 7.3

Cancel the common factor of $4$ .

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Step 7.3.1

Cancel the common factor.

$\frac{3 \cdot 4^{4 (\frac{1}{4})}}{2 \cdot 81^{\frac{1}{4}}}$

Step 7.3.2

Rewrite the expression.

$\frac{3 \cdot 4^{1}}{2 \cdot 81^{\frac{1}{4}}}$

Step 7.4

Evaluate the exponent.

$\frac{3 \cdot 4}{2 \cdot 81^{\frac{1}{4}}}$

Step 8

Simplify the denominator.

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Step 8.1

Rewrite $81$ as $3^{4}$ .

$\frac{3 \cdot 4}{2 \cdot {(3^{4})}^{\frac{1}{4}}}$

Step 8.2

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

$\frac{3 \cdot 4}{2 \cdot 3^{4 (\frac{1}{4})}}$

Step 8.3

Cancel the common factor of $4$ .

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Step 8.3.1

Cancel the common factor.

$\frac{3 \cdot 4}{2 \cdot 3^{4 (\frac{1}{4})}}$

Step 8.3.2

Rewrite the expression.

$\frac{3 \cdot 4}{2 \cdot 3^{1}}$

Step 8.4

Evaluate the exponent.

$\frac{3 \cdot 4}{2 \cdot 3}$

Step 9

Simplify the expression.

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Step 9.1

Multiply $3$ by $4$ .

$\frac{12}{2 \cdot 3}$

Step 9.2

Multiply $2$ by $3$ .

$\frac{12}{6}$

Step 9.3

Divide $12$ by $6$ .

$2$