Basic Math Examples

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

Step 1

Combine.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

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

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

Step 4.5.2

Multiply $2$ by $3$ .

$a^{- \frac{3}{6} + \frac{2}{3} \cdot \frac{2}{2}} b^{\frac{3}{2}} b^{- \frac{1}{2}}$

Step 4.5.3

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

$a^{- \frac{3}{6} + \frac{2 \cdot 2}{3 \cdot 2}} b^{\frac{3}{2}} b^{- \frac{1}{2}}$

Step 4.5.4

Multiply $3$ by $2$ .

$a^{- \frac{3}{6} + \frac{2 \cdot 2}{6}} b^{\frac{3}{2}} b^{- \frac{1}{2}}$

Step 4.6

Combine the numerators over the common denominator.

$a^{\frac{- 3 + 2 \cdot 2}{6}} b^{\frac{3}{2}} b^{- \frac{1}{2}}$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $2$ by $2$ .

$a^{\frac{- 3 + 4}{6}} b^{\frac{3}{2}} b^{- \frac{1}{2}}$

Step 4.7.2

Add $- 3$ and $4$ .

$a^{\frac{1}{6}} b^{\frac{3}{2}} b^{- \frac{1}{2}}$

Step 5

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

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Step 5.1

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

$a^{\frac{1}{6}} (b^{- \frac{1}{2}} b^{\frac{3}{2}})$

Step 5.2

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

$a^{\frac{1}{6}} b^{- \frac{1}{2} + \frac{3}{2}}$

Step 5.3

Combine the numerators over the common denominator.

$a^{\frac{1}{6}} b^{\frac{- 1 + 3}{2}}$

Step 5.4

Add $- 1$ and $3$ .

$a^{\frac{1}{6}} b^{\frac{2}{2}}$

Step 5.5

Divide $2$ by $2$ .

$a^{\frac{1}{6}} b^{1}$

Step 6

Simplify $a^{\frac{1}{6}} b^{1}$ .

$a^{\frac{1}{6}} b$

Step 7

Reorder factors in $a^{\frac{1}{6}} b$ .

$b a^{\frac{1}{6}}$