Algebra Examples

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

Step 1

Move $y^{\frac{2}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 2

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

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Step 2.1

Move $y^{- \frac{2}{3}}$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

Multiply $3$ by $4$ .

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

Step 2.5.3

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

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

Step 2.5.4

Multiply $4$ by $3$ .

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 2$ by $4$ .

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

Step 2.7.2

Multiply $3$ by $3$ .

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

Step 2.7.3

Add $- 8$ and $9$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

Move $x^{- \frac{1}{3}}$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

Multiply $3$ by $2$ .

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

Step 4.5.3

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

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

Step 4.5.4

Multiply $2$ by $3$ .

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

Step 4.6

Combine the numerators over the common denominator.

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

Step 4.7

Add $- 2$ and $3$ .

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

Step 5

Move the negative in front of the fraction.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Simplify the expression.

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Step 7.1

Raise $- 1$ to the power of $6$ .

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

Step 7.2

Multiply $\frac{{(x^{\frac{1}{6}})}^{6}}{{(y^{\frac{1}{12}})}^{6}}$ by $1$ .

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

Step 8

Simplify the numerator.

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Step 8.1

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

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Step 8.1.1

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

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

Step 8.1.2

Cancel the common factor of $6$ .

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Step 8.1.2.1

Cancel the common factor.

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

Step 8.1.2.2

Rewrite the expression.

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

Step 8.2

Simplify.

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

Step 9

Multiply the exponents in ${(y^{\frac{1}{12}})}^{6}$ .

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Step 9.1

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

$\frac{x}{y^{\frac{1}{12} \cdot 6}}$

Step 9.2

Cancel the common factor of $6$ .

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Step 9.2.1

Factor $6$ out of $12$ .

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

Step 9.2.2

Cancel the common factor.

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

Step 9.2.3

Rewrite the expression.

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