Precalculus Examples

$\frac{a^{\frac{3}{5}} a^{\frac{1}{3}}}{a^{\frac{3}{2}}}$

Step 1

Multiply $a^{\frac{3}{5}}$ by $a^{\frac{1}{3}}$ by adding the exponents.

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Step 1.1

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

$\frac{a^{\frac{3}{5} + \frac{1}{3}}}{a^{\frac{3}{2}}}$

Step 1.2

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

$\frac{a^{\frac{3}{5} \cdot \frac{3}{3} + \frac{1}{3}}}{a^{\frac{3}{2}}}$

Step 1.3

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

$\frac{a^{\frac{3}{5} \cdot \frac{3}{3} + \frac{1}{3} \cdot \frac{5}{5}}}{a^{\frac{3}{2}}}$

Step 1.4

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

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Step 1.4.1

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

$\frac{a^{\frac{3 \cdot 3}{5 \cdot 3} + \frac{1}{3} \cdot \frac{5}{5}}}{a^{\frac{3}{2}}}$

Step 1.4.2

Multiply $5$ by $3$ .

$\frac{a^{\frac{3 \cdot 3}{15} + \frac{1}{3} \cdot \frac{5}{5}}}{a^{\frac{3}{2}}}$

Step 1.4.3

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

$\frac{a^{\frac{3 \cdot 3}{15} + \frac{5}{3 \cdot 5}}}{a^{\frac{3}{2}}}$

Step 1.4.4

Multiply $3$ by $5$ .

$\frac{a^{\frac{3 \cdot 3}{15} + \frac{5}{15}}}{a^{\frac{3}{2}}}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{a^{\frac{3 \cdot 3 + 5}{15}}}{a^{\frac{3}{2}}}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $3$ by $3$ .

$\frac{a^{\frac{9 + 5}{15}}}{a^{\frac{3}{2}}}$

Step 1.6.2

Add $9$ and $5$ .

$\frac{a^{\frac{14}{15}}}{a^{\frac{3}{2}}}$

Step 2

Move $a^{\frac{14}{15}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{1}{a^{\frac{3}{2}} a^{- \frac{14}{15}}}$

Step 3

Multiply $a^{\frac{3}{2}}$ by $a^{- \frac{14}{15}}$ by adding the exponents.

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Step 3.1

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

$\frac{1}{a^{\frac{3}{2} - \frac{14}{15}}}$

Step 3.2

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\frac{1}{a^{\frac{3}{2} \cdot \frac{15}{15} - \frac{14}{15}}}$

Step 3.3

To write $- \frac{14}{15}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{a^{\frac{3}{2} \cdot \frac{15}{15} - \frac{14}{15} \cdot \frac{2}{2}}}$

Step 3.4

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{3}{2}$ by $\frac{15}{15}$ .

$\frac{1}{a^{\frac{3 \cdot 15}{2 \cdot 15} - \frac{14}{15} \cdot \frac{2}{2}}}$

Step 3.4.2

Multiply $2$ by $15$ .

$\frac{1}{a^{\frac{3 \cdot 15}{30} - \frac{14}{15} \cdot \frac{2}{2}}}$

Step 3.4.3

Multiply $\frac{14}{15}$ by $\frac{2}{2}$ .

$\frac{1}{a^{\frac{3 \cdot 15}{30} - \frac{14 \cdot 2}{15 \cdot 2}}}$

Step 3.4.4

Multiply $15$ by $2$ .

$\frac{1}{a^{\frac{3 \cdot 15}{30} - \frac{14 \cdot 2}{30}}}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{1}{a^{\frac{3 \cdot 15 - 14 \cdot 2}{30}}}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $3$ by $15$ .

$\frac{1}{a^{\frac{45 - 14 \cdot 2}{30}}}$

Step 3.6.2

Multiply $- 14$ by $2$ .

$\frac{1}{a^{\frac{45 - 28}{30}}}$

Step 3.6.3

Subtract $28$ from $45$ .

$\frac{1}{a^{\frac{17}{30}}}$