Algebra Examples

${(\frac{a^{\frac{2}{3}} b}{a b^{\frac{3}{4}}})}^{3}$

Step 1

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

${(\frac{b}{a b^{\frac{3}{4}} a^{- \frac{2}{3}}})}^{3}$

Step 2

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

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Step 2.1

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

${(\frac{b}{a^{- \frac{2}{3}} a b^{\frac{3}{4}}})}^{3}$

Step 2.2

Multiply $a^{- \frac{2}{3}}$ by $a$ .

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Step 2.2.1

Raise $a$ to the power of $1$ .

${(\frac{b}{a^{- \frac{2}{3}} a^{1} b^{\frac{3}{4}}})}^{3}$

Step 2.2.2

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

${(\frac{b}{a^{- \frac{2}{3} + 1} b^{\frac{3}{4}}})}^{3}$

Step 2.3

Write $1$ as a fraction with a common denominator.

${(\frac{b}{a^{- \frac{2}{3} + \frac{3}{3}} b^{\frac{3}{4}}})}^{3}$

Step 2.4

Combine the numerators over the common denominator.

${(\frac{b}{a^{\frac{- 2 + 3}{3}} b^{\frac{3}{4}}})}^{3}$

Step 2.5

Add $- 2$ and $3$ .

${(\frac{b}{a^{\frac{1}{3}} b^{\frac{3}{4}}})}^{3}$

Step 3

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

${(\frac{b \cdot b^{- \frac{3}{4}}}{a^{\frac{1}{3}}})}^{3}$

Step 4

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

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Step 4.1

Multiply $b$ by $b^{- \frac{3}{4}}$ .

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Step 4.1.1

Raise $b$ to the power of $1$ .

${(\frac{b^{1} b^{- \frac{3}{4}}}{a^{\frac{1}{3}}})}^{3}$

Step 4.1.2

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

${(\frac{b^{1 - \frac{3}{4}}}{a^{\frac{1}{3}}})}^{3}$

Step 4.2

Write $1$ as a fraction with a common denominator.

${(\frac{b^{\frac{4}{4} - \frac{3}{4}}}{a^{\frac{1}{3}}})}^{3}$

Step 4.3

Combine the numerators over the common denominator.

${(\frac{b^{\frac{4 - 3}{4}}}{a^{\frac{1}{3}}})}^{3}$

Step 4.4

Subtract $3$ from $4$ .

${(\frac{b^{\frac{1}{4}}}{a^{\frac{1}{3}}})}^{3}$

Step 5

Apply basic rules of exponents.

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Step 5.1

Apply the product rule to $\frac{b^{\frac{1}{4}}}{a^{\frac{1}{3}}}$ .

$\frac{{(b^{\frac{1}{4}})}^{3}}{{(a^{\frac{1}{3}})}^{3}}$

Step 5.2

Multiply the exponents in ${(b^{\frac{1}{4}})}^{3}$ .

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Step 5.2.1

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

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

Step 5.2.2

Combine $\frac{1}{4}$ and $3$ .

$\frac{b^{\frac{3}{4}}}{{(a^{\frac{1}{3}})}^{3}}$

Step 6

Simplify the denominator.

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Step 6.1

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

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Step 6.1.1

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

$\frac{b^{\frac{3}{4}}}{a^{\frac{1}{3} \cdot 3}}$

Step 6.1.2

Cancel the common factor of $3$ .

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Step 6.1.2.1

Cancel the common factor.

$\frac{b^{\frac{3}{4}}}{a^{\frac{1}{3} \cdot 3}}$

Step 6.1.2.2

Rewrite the expression.

$\frac{b^{\frac{3}{4}}}{a^{1}}$

Step 6.2

Simplify.

$\frac{b^{\frac{3}{4}}}{a}$