Algebra Examples

${(\frac{x^{- \frac{1}{2}} y^{\frac{4}{5}}}{x^{\frac{2}{3}} y^{\frac{1}{3}}})}^{3}$

Step 1

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

${(\frac{y^{\frac{4}{5}}}{x^{\frac{2}{3}} y^{\frac{1}{3}} x^{\frac{1}{2}}})}^{3}$

Step 2

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

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Step 2.1

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

${(\frac{y^{\frac{4}{5}}}{x^{\frac{1}{2}} x^{\frac{2}{3}} y^{\frac{1}{3}}})}^{3}$

Step 2.2

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

${(\frac{y^{\frac{4}{5}}}{x^{\frac{1}{2} + \frac{2}{3}} y^{\frac{1}{3}}})}^{3}$

Step 2.3

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

${(\frac{y^{\frac{4}{5}}}{x^{\frac{1}{2} \cdot \frac{3}{3} + \frac{2}{3}} y^{\frac{1}{3}}})}^{3}$

Step 2.4

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

${(\frac{y^{\frac{4}{5}}}{x^{\frac{1}{2} \cdot \frac{3}{3} + \frac{2}{3} \cdot \frac{2}{2}} y^{\frac{1}{3}}})}^{3}$

Step 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 2.5.1

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

${(\frac{y^{\frac{4}{5}}}{x^{\frac{3}{2 \cdot 3} + \frac{2}{3} \cdot \frac{2}{2}} y^{\frac{1}{3}}})}^{3}$

Step 2.5.2

Multiply $2$ by $3$ .

${(\frac{y^{\frac{4}{5}}}{x^{\frac{3}{6} + \frac{2}{3} \cdot \frac{2}{2}} y^{\frac{1}{3}}})}^{3}$

Step 2.5.3

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

${(\frac{y^{\frac{4}{5}}}{x^{\frac{3}{6} + \frac{2 \cdot 2}{3 \cdot 2}} y^{\frac{1}{3}}})}^{3}$

Step 2.5.4

Multiply $3$ by $2$ .

${(\frac{y^{\frac{4}{5}}}{x^{\frac{3}{6} + \frac{2 \cdot 2}{6}} y^{\frac{1}{3}}})}^{3}$

Step 2.6

Combine the numerators over the common denominator.

${(\frac{y^{\frac{4}{5}}}{x^{\frac{3 + 2 \cdot 2}{6}} y^{\frac{1}{3}}})}^{3}$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $2$ by $2$ .

${(\frac{y^{\frac{4}{5}}}{x^{\frac{3 + 4}{6}} y^{\frac{1}{3}}})}^{3}$

Step 2.7.2

Add $3$ and $4$ .

${(\frac{y^{\frac{4}{5}}}{x^{\frac{7}{6}} y^{\frac{1}{3}}})}^{3}$

Step 3

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

${(\frac{y^{\frac{4}{5}} y^{- \frac{1}{3}}}{x^{\frac{7}{6}}})}^{3}$

Step 4

Multiply $y^{\frac{4}{5}}$ by $y^{- \frac{1}{3}}$ by adding the exponents.

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Step 4.1

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

${(\frac{y^{\frac{4}{5} - \frac{1}{3}}}{x^{\frac{7}{6}}})}^{3}$

Step 4.2

To write $\frac{4}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${(\frac{y^{\frac{4}{5} \cdot \frac{3}{3} - \frac{1}{3}}}{x^{\frac{7}{6}}})}^{3}$

Step 4.3

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

${(\frac{y^{\frac{4}{5} \cdot \frac{3}{3} - \frac{1}{3} \cdot \frac{5}{5}}}{x^{\frac{7}{6}}})}^{3}$

Step 4.4

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.1

Multiply $\frac{4}{5}$ by $\frac{3}{3}$ .

${(\frac{y^{\frac{4 \cdot 3}{5 \cdot 3} - \frac{1}{3} \cdot \frac{5}{5}}}{x^{\frac{7}{6}}})}^{3}$

Step 4.4.2

Multiply $5$ by $3$ .

${(\frac{y^{\frac{4 \cdot 3}{15} - \frac{1}{3} \cdot \frac{5}{5}}}{x^{\frac{7}{6}}})}^{3}$

Step 4.4.3

Multiply $\frac{1}{3}$ by $\frac{5}{5}$ .

${(\frac{y^{\frac{4 \cdot 3}{15} - \frac{5}{3 \cdot 5}}}{x^{\frac{7}{6}}})}^{3}$

Step 4.4.4

Multiply $3$ by $5$ .

${(\frac{y^{\frac{4 \cdot 3}{15} - \frac{5}{15}}}{x^{\frac{7}{6}}})}^{3}$

Step 4.5

Combine the numerators over the common denominator.

${(\frac{y^{\frac{4 \cdot 3 - 5}{15}}}{x^{\frac{7}{6}}})}^{3}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $4$ by $3$ .

${(\frac{y^{\frac{12 - 5}{15}}}{x^{\frac{7}{6}}})}^{3}$

Step 4.6.2

Subtract $5$ from $12$ .

${(\frac{y^{\frac{7}{15}}}{x^{\frac{7}{6}}})}^{3}$

Step 5

Apply the product rule to $\frac{y^{\frac{7}{15}}}{x^{\frac{7}{6}}}$ .

$\frac{{(y^{\frac{7}{15}})}^{3}}{{(x^{\frac{7}{6}})}^{3}}$

Step 6

Multiply the exponents in ${(y^{\frac{7}{15}})}^{3}$ .

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Step 6.1

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

$\frac{y^{\frac{7}{15} \cdot 3}}{{(x^{\frac{7}{6}})}^{3}}$

Step 6.2

Cancel the common factor of $3$ .

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Step 6.2.1

Factor $3$ out of $15$ .

$\frac{y^{\frac{7}{3 (5)} \cdot 3}}{{(x^{\frac{7}{6}})}^{3}}$

Step 6.2.2

Cancel the common factor.

$\frac{y^{\frac{7}{3 \cdot 5} \cdot 3}}{{(x^{\frac{7}{6}})}^{3}}$

Step 6.2.3

Rewrite the expression.

$\frac{y^{\frac{7}{5}}}{{(x^{\frac{7}{6}})}^{3}}$

Step 7

Multiply the exponents in ${(x^{\frac{7}{6}})}^{3}$ .

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Step 7.1

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

$\frac{y^{\frac{7}{5}}}{x^{\frac{7}{6} \cdot 3}}$

Step 7.2

Cancel the common factor of $3$ .

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Step 7.2.1

Factor $3$ out of $6$ .

$\frac{y^{\frac{7}{5}}}{x^{\frac{7}{3 (2)} \cdot 3}}$

Step 7.2.2

Cancel the common factor.

$\frac{y^{\frac{7}{5}}}{x^{\frac{7}{3 \cdot 2} \cdot 3}}$

Step 7.2.3

Rewrite the expression.

$\frac{y^{\frac{7}{5}}}{x^{\frac{7}{2}}}$