Basic Math Examples

$\sqrt{1 - \frac{8}{9} \cdot {(- 3)}^{2}} + {(- \frac{1}{2})}^{3} \div \frac{3}{1}$

Step 1

Simplify each term.

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Step 1.1

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

$\sqrt{1 - \frac{8}{9} \cdot 9} + {(- \frac{1}{2})}^{3} \div \frac{3}{1}$

Step 1.2

Cancel the common factor of $9$ .

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Step 1.2.1

Move the leading negative in $- \frac{8}{9}$ into the numerator.

$\sqrt{1 + \frac{- 8}{9} \cdot 9} + {(- \frac{1}{2})}^{3} \div \frac{3}{1}$

Step 1.2.2

Cancel the common factor.

$\sqrt{1 + \frac{- 8}{9} \cdot 9} + {(- \frac{1}{2})}^{3} \div \frac{3}{1}$

Step 1.2.3

Rewrite the expression.

$\sqrt{1 - 8} + {(- \frac{1}{2})}^{3} \div \frac{3}{1}$

Step 1.3

Subtract $8$ from $1$ .

$\sqrt{- 7} + {(- \frac{1}{2})}^{3} \div \frac{3}{1}$

Step 1.4

Rewrite $- 7$ as $- 1 (7)$ .

$\sqrt{- 1 (7)} + {(- \frac{1}{2})}^{3} \div \frac{3}{1}$

Step 1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$\sqrt{- 1} \cdot \sqrt{7} + {(- \frac{1}{2})}^{3} \div \frac{3}{1}$

Step 1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$i \cdot \sqrt{7} + {(- \frac{1}{2})}^{3} \div \frac{3}{1}$

Step 1.7

To divide by a fraction, multiply by its reciprocal.

$i \sqrt{7} + {(- \frac{1}{2})}^{3} \frac{1}{3}$

Step 1.8

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

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Step 1.8.1

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

$i \sqrt{7} + {(- 1)}^{3} {(\frac{1}{2})}^{3} \frac{1}{3}$

Step 1.8.2

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

$i \sqrt{7} + {(- 1)}^{3} \frac{1^{3}}{2^{3}} \frac{1}{3}$

Step 1.9

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

$i \sqrt{7} - \frac{1^{3}}{2^{3}} \cdot \frac{1}{3}$

Step 1.10

One to any power is one.

$i \sqrt{7} - \frac{1}{2^{3}} \cdot \frac{1}{3}$

Step 1.11

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

$i \sqrt{7} - \frac{1}{8} \cdot \frac{1}{3}$

Step 1.12

Multiply $- \frac{1}{8} \cdot \frac{1}{3}$ .

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Step 1.12.1

Multiply $\frac{1}{3}$ by $\frac{1}{8}$ .

$i \sqrt{7} - \frac{1}{3 \cdot 8}$

Step 1.12.2

Multiply $3$ by $8$ .

$i \sqrt{7} - \frac{1}{24}$

Step 2

To write $i \sqrt{7}$ as a fraction with a common denominator, multiply by $\frac{24}{24}$ .

$i \sqrt{7} \cdot \frac{24}{24} - \frac{1}{24}$

Step 3

Combine $i \sqrt{7}$ and $\frac{24}{24}$ .

$\frac{i \sqrt{7} \cdot 24}{24} - \frac{1}{24}$

Step 4

Combine the numerators over the common denominator.

$\frac{- 1 + 24 i \sqrt{7}}{24}$

Step 5

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- 1 (1) + 24 i \sqrt{7}}{24}$

Step 6

Factor $- 1$ out of $24 i \sqrt{7}$ .

$\frac{- 1 (1) - (- 24 i \sqrt{7})}{24}$

Step 7

Factor $- 1$ out of $- 1 (1) - (- 24 i \sqrt{7})$ .

$\frac{- 1 (1 - 24 i \sqrt{7})}{24}$

Step 8

Move the negative in front of the fraction.

$- \frac{1 - 24 i \sqrt{7}}{24}$