Precalculus Examples

${(4 b)}^{\frac{1}{2}} (8 b^{\frac{1}{4}})$

Step 1

Rewrite using the commutative property of multiplication.

$8 {(4 b)}^{\frac{1}{2}} b^{\frac{1}{4}}$

Step 2

Apply the product rule to $4 b$ .

$8 (4^{\frac{1}{2}} b^{\frac{1}{2}}) b^{\frac{1}{4}}$

Step 3

Multiply $b^{\frac{1}{2}}$ by $b^{\frac{1}{4}}$ by adding the exponents.

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Step 3.1

Move $b^{\frac{1}{4}}$ .

$8 (4^{\frac{1}{2}} (b^{\frac{1}{4}} b^{\frac{1}{2}}))$

Step 3.2

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

$8 (4^{\frac{1}{2}} b^{\frac{1}{4} + \frac{1}{2}})$

Step 3.3

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

$8 (4^{\frac{1}{2}} b^{\frac{1}{4} + \frac{1}{2} \cdot \frac{2}{2}})$

Step 3.4

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

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

$8 (4^{\frac{1}{2}} b^{\frac{1}{4} + \frac{2}{2 \cdot 2}})$

Step 3.4.2

Multiply $2$ by $2$ .

$8 (4^{\frac{1}{2}} b^{\frac{1}{4} + \frac{2}{4}})$

Step 3.5

Combine the numerators over the common denominator.

$8 (4^{\frac{1}{2}} b^{\frac{1 + 2}{4}})$

Step 3.6

Add $1$ and $2$ .

$8 (4^{\frac{1}{2}} b^{\frac{3}{4}})$

Step 4

Simplify the expression.

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Step 4.1

Rewrite $4$ as $2^{2}$ .

$8 ({(2^{2})}^{\frac{1}{2}} b^{\frac{3}{4}})$

Step 4.2

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

$8 (2^{2 (\frac{1}{2})} b^{\frac{3}{4}})$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Cancel the common factor.

$8 (2^{2 (\frac{1}{2})} b^{\frac{3}{4}})$

Step 5.2

Rewrite the expression.

$8 (2^{1} b^{\frac{3}{4}})$

Step 6

Evaluate the exponent.

$8 (2 b^{\frac{3}{4}})$

Step 7

Multiply $2$ by $8$ .

$16 b^{\frac{3}{4}}$