Calculus Examples

$(32^{\frac{3}{2}}) {(\frac{1}{2})}^{\frac{3}{2}}$

Step 1

Simplify the expression.

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Step 1.1

Apply the product rule to $\frac{1}{2}$ .

$32^{\frac{3}{2}} \frac{1^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

Step 1.2

One to any power is one.

$32^{\frac{3}{2}} \frac{1}{2^{\frac{3}{2}}}$

Step 2

Combine $32^{\frac{3}{2}}$ and $\frac{1}{2^{\frac{3}{2}}}$ .

$\frac{32^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

Step 3

Use the power of quotient rule $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

${(\frac{32}{2})}^{\frac{3}{2}}$

Step 4

Simplify the expression.

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Step 4.1

Divide $32$ by $2$ .

$16^{\frac{3}{2}}$

Step 4.2

Rewrite $16$ as $4^{2}$ .

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

Step 4.3

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

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

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$4^{3}$

Step 6

Raise $4$ to the power of $3$ .

$64$