Evaluate ( cube root of -1/8)^3+3 3/4

${(\sqrt[3]{- \frac{1}{8}})}^{3} + 3 \frac{3}{4}$

Step 1

Convert $3 \frac{3}{4}$ to an improper fraction.

A mixed number is an addition of its whole and fractional parts.

${(\sqrt[3]{- \frac{1}{8}})}^{3} + 3 + \frac{3}{4}$

Add $3$ and $\frac{3}{4}$ .

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

${(\sqrt[3]{- \frac{1}{8}})}^{3} + 3 \cdot \frac{4}{4} + \frac{3}{4}$

Combine $3$ and $\frac{4}{4}$ .

${(\sqrt[3]{- \frac{1}{8}})}^{3} + \frac{3 \cdot 4}{4} + \frac{3}{4}$

Combine the numerators over the common denominator.

${(\sqrt[3]{- \frac{1}{8}})}^{3} + \frac{3 \cdot 4 + 3}{4}$

Simplify the numerator.

Multiply $3$ by $4$ .

${(\sqrt[3]{- \frac{1}{8}})}^{3} + \frac{12 + 3}{4}$

Add $12$ and $3$ .

${(\sqrt[3]{- \frac{1}{8}})}^{3} + \frac{15}{4}$

Step 2

Simplify each term.

Rewrite $- \frac{1}{8}$ as ${(- \frac{1}{2})}^{3}$ .

${\sqrt[3]{{(- \frac{1}{2})}^{3}}}^{3} + \frac{15}{4}$

Pull terms out from under the radical, assuming real numbers.

${(- \frac{1}{2})}^{3} + \frac{15}{4}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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

${(- 1)}^{3} {(\frac{1}{2})}^{3} + \frac{15}{4}$

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

${(- 1)}^{3} \frac{1^{3}}{2^{3}} + \frac{15}{4}$

Raise $- 1$ to the power of $3$ .

$- \frac{1^{3}}{2^{3}} + \frac{15}{4}$

One to any power is one.

$- \frac{1}{2^{3}} + \frac{15}{4}$

Raise $2$ to the power of $3$ .

$- \frac{1}{8} + \frac{15}{4}$

Step 3

To write $\frac{15}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{8} + \frac{15}{4} \cdot \frac{2}{2}$

Step 4

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{15}{4}$ by $\frac{2}{2}$ .

$- \frac{1}{8} + \frac{15 \cdot 2}{4 \cdot 2}$

Multiply $4$ by $2$ .

$- \frac{1}{8} + \frac{15 \cdot 2}{8}$

Step 5

Combine the numerators over the common denominator.

$\frac{- 1 + 15 \cdot 2}{8}$

Step 6

Simplify the numerator.

Multiply $15$ by $2$ .

$\frac{- 1 + 30}{8}$

Add $- 1$ and $30$ .

$\frac{29}{8}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{29}{8}$

Decimal Form:

$3.625$

Mixed Number Form:

$3 \frac{5}{8}$

Evaluate ( cube root of -1/8)^3+3 3/4

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