Precalculus Examples

${(\frac{x^{- \frac{3}{4}} y^{- \frac{1}{2}}}{x^{\frac{2}{3}} y^{\frac{2}{3}}})}^{6}$

Step 1

Move $x^{- \frac{3}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{y^{- \frac{1}{2}}}{x^{\frac{2}{3}} y^{\frac{2}{3}} x^{\frac{3}{4}}})}^{6}$

Step 2

Move $y^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{1}{x^{\frac{2}{3}} y^{\frac{2}{3}} x^{\frac{3}{4}} y^{\frac{1}{2}}})}^{6}$

Step 3

Simplify the denominator.

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Step 3.1

Multiply $x^{\frac{2}{3}}$ by $x^{\frac{3}{4}}$ by adding the exponents.

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Step 3.1.1

Move $x^{\frac{3}{4}}$ .

${(\frac{1}{x^{\frac{3}{4}} x^{\frac{2}{3}} y^{\frac{2}{3}} y^{\frac{1}{2}}})}^{6}$

Step 3.1.2

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

${(\frac{1}{x^{\frac{3}{4} + \frac{2}{3}} y^{\frac{2}{3}} y^{\frac{1}{2}}})}^{6}$

Step 3.1.3

To write $\frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${(\frac{1}{x^{\frac{3}{4} \cdot \frac{3}{3} + \frac{2}{3}} y^{\frac{2}{3}} y^{\frac{1}{2}}})}^{6}$

Step 3.1.4

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

${(\frac{1}{x^{\frac{3}{4} \cdot \frac{3}{3} + \frac{2}{3} \cdot \frac{4}{4}} y^{\frac{2}{3}} y^{\frac{1}{2}}})}^{6}$

Step 3.1.5

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1.5.1

Multiply $\frac{3}{4}$ by $\frac{3}{3}$ .

${(\frac{1}{x^{\frac{3 \cdot 3}{4 \cdot 3} + \frac{2}{3} \cdot \frac{4}{4}} y^{\frac{2}{3}} y^{\frac{1}{2}}})}^{6}$

Step 3.1.5.2

Multiply $4$ by $3$ .

${(\frac{1}{x^{\frac{3 \cdot 3}{12} + \frac{2}{3} \cdot \frac{4}{4}} y^{\frac{2}{3}} y^{\frac{1}{2}}})}^{6}$

Step 3.1.5.3

Multiply $\frac{2}{3}$ by $\frac{4}{4}$ .

${(\frac{1}{x^{\frac{3 \cdot 3}{12} + \frac{2 \cdot 4}{3 \cdot 4}} y^{\frac{2}{3}} y^{\frac{1}{2}}})}^{6}$

Step 3.1.5.4

Multiply $3$ by $4$ .

${(\frac{1}{x^{\frac{3 \cdot 3}{12} + \frac{2 \cdot 4}{12}} y^{\frac{2}{3}} y^{\frac{1}{2}}})}^{6}$

Step 3.1.6

Combine the numerators over the common denominator.

${(\frac{1}{x^{\frac{3 \cdot 3 + 2 \cdot 4}{12}} y^{\frac{2}{3}} y^{\frac{1}{2}}})}^{6}$

Step 3.1.7

Simplify the numerator.

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Step 3.1.7.1

Multiply $3$ by $3$ .

${(\frac{1}{x^{\frac{9 + 2 \cdot 4}{12}} y^{\frac{2}{3}} y^{\frac{1}{2}}})}^{6}$

Step 3.1.7.2

Multiply $2$ by $4$ .

${(\frac{1}{x^{\frac{9 + 8}{12}} y^{\frac{2}{3}} y^{\frac{1}{2}}})}^{6}$

Step 3.1.7.3

Add $9$ and $8$ .

${(\frac{1}{x^{\frac{17}{12}} y^{\frac{2}{3}} y^{\frac{1}{2}}})}^{6}$

Step 3.2

Multiply $y^{\frac{2}{3}}$ by $y^{\frac{1}{2}}$ by adding the exponents.

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Step 3.2.1

Move $y^{\frac{1}{2}}$ .

${(\frac{1}{x^{\frac{17}{12}} (y^{\frac{1}{2}} y^{\frac{2}{3}})})}^{6}$

Step 3.2.2

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

${(\frac{1}{x^{\frac{17}{12}} y^{\frac{1}{2} + \frac{2}{3}}})}^{6}$

Step 3.2.3

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${(\frac{1}{x^{\frac{17}{12}} y^{\frac{1}{2} \cdot \frac{3}{3} + \frac{2}{3}}})}^{6}$

Step 3.2.4

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(\frac{1}{x^{\frac{17}{12}} y^{\frac{1}{2} \cdot \frac{3}{3} + \frac{2}{3} \cdot \frac{2}{2}}})}^{6}$

Step 3.2.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.5.1

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

${(\frac{1}{x^{\frac{17}{12}} y^{\frac{3}{2 \cdot 3} + \frac{2}{3} \cdot \frac{2}{2}}})}^{6}$

Step 3.2.5.2

Multiply $2$ by $3$ .

${(\frac{1}{x^{\frac{17}{12}} y^{\frac{3}{6} + \frac{2}{3} \cdot \frac{2}{2}}})}^{6}$

Step 3.2.5.3

Multiply $\frac{2}{3}$ by $\frac{2}{2}$ .

${(\frac{1}{x^{\frac{17}{12}} y^{\frac{3}{6} + \frac{2 \cdot 2}{3 \cdot 2}}})}^{6}$

Step 3.2.5.4

Multiply $3$ by $2$ .

${(\frac{1}{x^{\frac{17}{12}} y^{\frac{3}{6} + \frac{2 \cdot 2}{6}}})}^{6}$

Step 3.2.6

Combine the numerators over the common denominator.

${(\frac{1}{x^{\frac{17}{12}} y^{\frac{3 + 2 \cdot 2}{6}}})}^{6}$

Step 3.2.7

Simplify the numerator.

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Step 3.2.7.1

Multiply $2$ by $2$ .

${(\frac{1}{x^{\frac{17}{12}} y^{\frac{3 + 4}{6}}})}^{6}$

Step 3.2.7.2

Add $3$ and $4$ .

${(\frac{1}{x^{\frac{17}{12}} y^{\frac{7}{6}}})}^{6}$

Step 4

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

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Step 4.1

Apply the product rule to $\frac{1}{x^{\frac{17}{12}} y^{\frac{7}{6}}}$ .

$\frac{1^{6}}{{(x^{\frac{17}{12}} y^{\frac{7}{6}})}^{6}}$

Step 4.2

Apply the product rule to $x^{\frac{17}{12}} y^{\frac{7}{6}}$ .

$\frac{1^{6}}{{(x^{\frac{17}{12}})}^{6} {(y^{\frac{7}{6}})}^{6}}$

Step 5

One to any power is one.

$\frac{1}{{(x^{\frac{17}{12}})}^{6} {(y^{\frac{7}{6}})}^{6}}$

Step 6

Simplify the denominator.

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Step 6.1

Multiply the exponents in ${(x^{\frac{17}{12}})}^{6}$ .

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Step 6.1.1

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

$\frac{1}{x^{\frac{17}{12} \cdot 6} {(y^{\frac{7}{6}})}^{6}}$

Step 6.1.2

Cancel the common factor of $6$ .

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Step 6.1.2.1

Factor $6$ out of $12$ .

$\frac{1}{x^{\frac{17}{6 (2)} \cdot 6} {(y^{\frac{7}{6}})}^{6}}$

Step 6.1.2.2

Cancel the common factor.

$\frac{1}{x^{\frac{17}{6 \cdot 2} \cdot 6} {(y^{\frac{7}{6}})}^{6}}$

Step 6.1.2.3

Rewrite the expression.

$\frac{1}{x^{\frac{17}{2}} {(y^{\frac{7}{6}})}^{6}}$

Step 6.2

Multiply the exponents in ${(y^{\frac{7}{6}})}^{6}$ .

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Step 6.2.1

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

$\frac{1}{x^{\frac{17}{2}} y^{\frac{7}{6} \cdot 6}}$

Step 6.2.2

Cancel the common factor of $6$ .

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Step 6.2.2.1

Cancel the common factor.

$\frac{1}{x^{\frac{17}{2}} y^{\frac{7}{6} \cdot 6}}$

Step 6.2.2.2

Rewrite the expression.

$\frac{1}{x^{\frac{17}{2}} y^{7}}$

Step 7

Reorder factors in $\frac{1}{x^{\frac{17}{2}} y^{7}}$ .

$\frac{1}{y^{7} x^{\frac{17}{2}}}$