Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Multiply $b^{- \frac{1}{3}}$ by $b$ .

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Add $- 1$ and $3$ .

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

Step 3

Multiply by $1$ .

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

Step 4

Cancel the common factors.

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Step 4.1

Factor $a^{2}$ out of $a^{3} b^{\frac{2}{3}}$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

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

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Step 5.1

Apply the product rule to $\frac{1}{a b^{\frac{2}{3}}}$ .

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

Step 5.2

Apply the product rule to $a b^{\frac{2}{3}}$ .

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

Step 6

One to any power is one.

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

Step 7

Multiply the exponents in ${(b^{\frac{2}{3}})}^{\frac{1}{2}}$ .

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Step 7.1

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

$\frac{1}{a^{\frac{1}{2}} b^{\frac{2}{3} \cdot \frac{1}{2}}}$

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Cancel the common factor.

$\frac{1}{a^{\frac{1}{2}} b^{\frac{2}{3} \cdot \frac{1}{2}}}$

Step 7.2.2

Rewrite the expression.

$\frac{1}{a^{\frac{1}{2}} b^{\frac{1}{3}}}$