Calculus Examples

$\frac{2}{3} \cdot (8^{\frac{3}{2}})$

Step 1

Combine $\frac{2}{3}$ and $8^{\frac{3}{2}}$ .

$\frac{2 \cdot 8^{\frac{3}{2}}}{3}$

Step 2

Rewrite $8$ as $2^{3}$ .

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

Step 3

Multiply the exponents in ${(2^{3})}^{\frac{3}{2}}$ .

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Step 3.1

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

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

Step 3.2

Multiply $3 (\frac{3}{2})$ .

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Step 3.2.1

Combine $3$ and $\frac{3}{2}$ .

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

Step 3.2.2

Multiply $3$ by $3$ .

$\frac{2 \cdot 2^{\frac{9}{2}}}{3}$

Step 4

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

$\frac{2^{1 + \frac{9}{2}}}{3}$

Step 5

Write $1$ as a fraction with a common denominator.

$\frac{2^{\frac{2}{2} + \frac{9}{2}}}{3}$

Step 6

Combine the numerators over the common denominator.

$\frac{2^{\frac{2 + 9}{2}}}{3}$

Step 7

Add $2$ and $9$ .

$\frac{2^{\frac{11}{2}}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{2^{\frac{11}{2}}}{3}$

Decimal Form:

$15.08494466 \dots$