Precalculus Examples

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

Step 1

Multiply $b^{\frac{1}{4}}$ by $b^{- \frac{3}{4}}$ 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.

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

Step 1.2

Combine the numerators over the common denominator.

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

Step 1.3

Subtract $3$ from $1$ .

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

Step 1.4

Cancel the common factor of $- 2$ and $4$ .

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Step 1.4.1

Factor $2$ out of $- 2$ .

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

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $2$ out of $4$ .

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

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

Step 1.5

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

Multiply $b^{\frac{1}{3}}$ by $b^{\frac{1}{2}}$ 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.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

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

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

Step 3.4.2

Multiply $3$ by $2$ .

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

Step 3.4.3

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

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

Step 3.4.4

Multiply $2$ by $3$ .

$b^{\frac{2}{6} + \frac{3}{6}}$

Step 3.5

Combine the numerators over the common denominator.

$b^{\frac{2 + 3}{6}}$

Step 3.6

Add $2$ and $3$ .

$b^{\frac{5}{6}}$