Algebra Examples

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

Step 1

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

$\frac{a^{\frac{2}{3}}}{b^{\frac{3}{2}} \cdot \sqrt[3]{a} b^{- \frac{1}{2}}}$

Step 2

Simplify the denominator.

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Step 2.1

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

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Step 2.1.1

Move $b^{- \frac{1}{2}}$ .

$\frac{a^{\frac{2}{3}}}{b^{- \frac{1}{2}} b^{\frac{3}{2}} \cdot \sqrt[3]{a}}$

Step 2.1.2

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

$\frac{a^{\frac{2}{3}}}{b^{- \frac{1}{2} + \frac{3}{2}} \cdot \sqrt[3]{a}}$

Step 2.1.3

Combine the numerators over the common denominator.

$\frac{a^{\frac{2}{3}}}{b^{\frac{- 1 + 3}{2}} \cdot \sqrt[3]{a}}$

Step 2.1.4

Add $- 1$ and $3$ .

$\frac{a^{\frac{2}{3}}}{b^{\frac{2}{2}} \cdot \sqrt[3]{a}}$

Step 2.1.5

Divide $2$ by $2$ .

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

Step 2.2

Simplify $b^{1} \cdot \sqrt[3]{a}$ .

$\frac{a^{\frac{2}{3}}}{b \cdot \sqrt[3]{a}}$

Step 3

Multiply $\frac{a^{\frac{2}{3}}}{b \cdot \sqrt[3]{a}}$ by $\frac{{\sqrt[3]{a}}^{2}}{{\sqrt[3]{a}}^{2}}$ .

$\frac{a^{\frac{2}{3}}}{b \cdot \sqrt[3]{a}} \cdot \frac{{\sqrt[3]{a}}^{2}}{{\sqrt[3]{a}}^{2}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{a^{\frac{2}{3}}}{b \cdot \sqrt[3]{a}}$ by $\frac{{\sqrt[3]{a}}^{2}}{{\sqrt[3]{a}}^{2}}$ .

$\frac{a^{\frac{2}{3}} {\sqrt[3]{a}}^{2}}{b \cdot \sqrt[3]{a} {\sqrt[3]{a}}^{2}}$

Step 4.2

Move $\sqrt[3]{a}$ .

$\frac{a^{\frac{2}{3}} {\sqrt[3]{a}}^{2}}{b (\sqrt[3]{a} {\sqrt[3]{a}}^{2})}$

Step 4.3

Raise $\sqrt[3]{a}$ to the power of $1$ .

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

Step 4.4

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

$\frac{a^{\frac{2}{3}} {\sqrt[3]{a}}^{2}}{b {\sqrt[3]{a}}^{1 + 2}}$

Step 4.5

Add $1$ and $2$ .

$\frac{a^{\frac{2}{3}} {\sqrt[3]{a}}^{2}}{b {\sqrt[3]{a}}^{3}}$

Step 4.6

Rewrite ${\sqrt[3]{a}}^{3}$ as $a$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{a}$ as $a^{\frac{1}{3}}$ .

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

Step 4.6.2

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

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

Step 4.6.3

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

$\frac{a^{\frac{2}{3}} {\sqrt[3]{a}}^{2}}{b a^{\frac{3}{3}}}$

Step 4.6.4

Cancel the common factor of $3$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{a^{\frac{2}{3}} {\sqrt[3]{a}}^{2}}{b a^{\frac{3}{3}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{a^{\frac{2}{3}} {\sqrt[3]{a}}^{2}}{b a^{1}}$

Step 4.6.5

Simplify.

$\frac{a^{\frac{2}{3}} {\sqrt[3]{a}}^{2}}{b a}$

Step 5

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

$\frac{{\sqrt[3]{a}}^{2}}{b a \cdot a^{- \frac{2}{3}}}$

Step 6

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

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Step 6.1

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

$\frac{{\sqrt[3]{a}}^{2}}{b (a^{- \frac{2}{3}} a)}$

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

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

$\frac{{\sqrt[3]{a}}^{2}}{b a^{- \frac{2}{3} + 1}}$

Step 6.3

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

$\frac{{\sqrt[3]{a}}^{2}}{b a^{- \frac{2}{3} + \frac{3}{3}}}$

Step 6.4

Combine the numerators over the common denominator.

$\frac{{\sqrt[3]{a}}^{2}}{b a^{\frac{- 2 + 3}{3}}}$

Step 6.5

Add $- 2$ and $3$ .

$\frac{{\sqrt[3]{a}}^{2}}{b a^{\frac{1}{3}}}$

Step 7

Rewrite ${\sqrt[3]{a}}^{2}$ as $\sqrt[3]{a^{2}}$ .

$\frac{\sqrt[3]{a^{2}}}{b a^{\frac{1}{3}}}$