Simplify (-2/3)^-2+(3/10)÷(1/4)- square root of (1/2+1/3)÷(10/3)

${(- \frac{2}{3})}^{- 2} + \frac{3}{10} \div \frac{1}{4} - \sqrt{(\frac{1}{2} + \frac{1}{3}) \div \frac{10}{3}}$

Step 1

Simplify each term.

Change the sign of the exponent by rewriting the base as its reciprocal.

${(- \frac{3}{2})}^{2} + \frac{3}{10} \div \frac{1}{4} - \sqrt{(\frac{1}{2} + \frac{1}{3}) \div \frac{10}{3}}$

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

Apply the product rule to $- \frac{3}{2}$ .

${(- 1)}^{2} {(\frac{3}{2})}^{2} + \frac{3}{10} \div \frac{1}{4} - \sqrt{(\frac{1}{2} + \frac{1}{3}) \div \frac{10}{3}}$

Apply the product rule to $\frac{3}{2}$ .

${(- 1)}^{2} \frac{3^{2}}{2^{2}} + \frac{3}{10} \div \frac{1}{4} - \sqrt{(\frac{1}{2} + \frac{1}{3}) \div \frac{10}{3}}$

Raise $- 1$ to the power of $2$ .

$1 \frac{3^{2}}{2^{2}} + \frac{3}{10} \div \frac{1}{4} - \sqrt{(\frac{1}{2} + \frac{1}{3}) \div \frac{10}{3}}$

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$\frac{3^{2}}{2^{2}} + \frac{3}{10} \div \frac{1}{4} - \sqrt{(\frac{1}{2} + \frac{1}{3}) \div \frac{10}{3}}$

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

$\frac{9}{2^{2}} + \frac{3}{10} \div \frac{1}{4} - \sqrt{(\frac{1}{2} + \frac{1}{3}) \div \frac{10}{3}}$

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

$\frac{9}{4} + \frac{3}{10} \div \frac{1}{4} - \sqrt{(\frac{1}{2} + \frac{1}{3}) \div \frac{10}{3}}$

To divide by a fraction, multiply by its reciprocal.

$\frac{9}{4} + \frac{3}{10} \cdot 4 - \sqrt{(\frac{1}{2} + \frac{1}{3}) \div \frac{10}{3}}$

Cancel the common factor of $2$ .

Factor $2$ out of $10$ .

$\frac{9}{4} + \frac{3}{2 (5)} \cdot 4 - \sqrt{(\frac{1}{2} + \frac{1}{3}) \div \frac{10}{3}}$

Factor $2$ out of $4$ .

$\frac{9}{4} + \frac{3}{2 \cdot 5} \cdot (2 \cdot 2) - \sqrt{(\frac{1}{2} + \frac{1}{3}) \div \frac{10}{3}}$

Cancel the common factor.

$\frac{9}{4} + \frac{3}{2 \cdot 5} \cdot (2 \cdot 2) - \sqrt{(\frac{1}{2} + \frac{1}{3}) \div \frac{10}{3}}$

Rewrite the expression.

$\frac{9}{4} + \frac{3}{5} \cdot 2 - \sqrt{(\frac{1}{2} + \frac{1}{3}) \div \frac{10}{3}}$

Combine $\frac{3}{5}$ and $2$ .

$\frac{9}{4} + \frac{3 \cdot 2}{5} - \sqrt{(\frac{1}{2} + \frac{1}{3}) \div \frac{10}{3}}$

Multiply $3$ by $2$ .

$\frac{9}{4} + \frac{6}{5} - \sqrt{(\frac{1}{2} + \frac{1}{3}) \div \frac{10}{3}}$

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{9}{4} + \frac{6}{5} - \sqrt{(\frac{1}{2} \cdot \frac{3}{3} + \frac{1}{3}) \div \frac{10}{3}}$

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{9}{4} + \frac{6}{5} - \sqrt{(\frac{1}{2} \cdot \frac{3}{3} + \frac{1}{3} \cdot \frac{2}{2}) \div \frac{10}{3}}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$\frac{9}{4} + \frac{6}{5} - \sqrt{(\frac{3}{2 \cdot 3} + \frac{1}{3} \cdot \frac{2}{2}) \div \frac{10}{3}}$

Multiply $2$ by $3$ .

$\frac{9}{4} + \frac{6}{5} - \sqrt{(\frac{3}{6} + \frac{1}{3} \cdot \frac{2}{2}) \div \frac{10}{3}}$

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$\frac{9}{4} + \frac{6}{5} - \sqrt{(\frac{3}{6} + \frac{2}{3 \cdot 2}) \div \frac{10}{3}}$

Multiply $3$ by $2$ .

$\frac{9}{4} + \frac{6}{5} - \sqrt{(\frac{3}{6} + \frac{2}{6}) \div \frac{10}{3}}$

Combine the numerators over the common denominator.

$\frac{9}{4} + \frac{6}{5} - \sqrt{\frac{3 + 2}{6} \div \frac{10}{3}}$

Add $3$ and $2$ .

$\frac{9}{4} + \frac{6}{5} - \sqrt{\frac{5}{6} \div \frac{10}{3}}$

To divide by a fraction, multiply by its reciprocal.

$\frac{9}{4} + \frac{6}{5} - \sqrt{\frac{5}{6} \cdot \frac{3}{10}}$

Cancel the common factor of $5$ .

Factor $5$ out of $10$ .

$\frac{9}{4} + \frac{6}{5} - \sqrt{\frac{5}{6} \cdot \frac{3}{5 (2)}}$

Cancel the common factor.

$\frac{9}{4} + \frac{6}{5} - \sqrt{\frac{5}{6} \cdot \frac{3}{5 \cdot 2}}$

Rewrite the expression.

$\frac{9}{4} + \frac{6}{5} - \sqrt{\frac{1}{6} \cdot \frac{3}{2}}$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$\frac{9}{4} + \frac{6}{5} - \sqrt{\frac{1}{3 (2)} \cdot \frac{3}{2}}$

Cancel the common factor.

$\frac{9}{4} + \frac{6}{5} - \sqrt{\frac{1}{3 \cdot 2} \cdot \frac{3}{2}}$

Rewrite the expression.

$\frac{9}{4} + \frac{6}{5} - \sqrt{\frac{1}{2} \cdot \frac{1}{2}}$

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

$\frac{9}{4} + \frac{6}{5} - \sqrt{\frac{1}{2 \cdot 2}}$

Multiply $2$ by $2$ .

$\frac{9}{4} + \frac{6}{5} - \sqrt{\frac{1}{4}}$

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$\frac{9}{4} + \frac{6}{5} - \frac{\sqrt{1}}{\sqrt{4}}$

Any root of $1$ is $1$ .

$\frac{9}{4} + \frac{6}{5} - \frac{1}{\sqrt{4}}$

Simplify the denominator.

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

$\frac{9}{4} + \frac{6}{5} - \frac{1}{\sqrt{2^{2}}}$

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

$\frac{9}{4} + \frac{6}{5} - \frac{1}{2}$

Step 2

Find the common denominator.

Multiply $\frac{9}{4}$ by $\frac{5}{5}$ .

$\frac{9}{4} \cdot \frac{5}{5} + \frac{6}{5} - \frac{1}{2}$

Multiply $\frac{9}{4}$ by $\frac{5}{5}$ .

$\frac{9 \cdot 5}{4 \cdot 5} + \frac{6}{5} - \frac{1}{2}$

Multiply $\frac{6}{5}$ by $\frac{4}{4}$ .

$\frac{9 \cdot 5}{4 \cdot 5} + \frac{6}{5} \cdot \frac{4}{4} - \frac{1}{2}$

Multiply $\frac{6}{5}$ by $\frac{4}{4}$ .

$\frac{9 \cdot 5}{4 \cdot 5} + \frac{6 \cdot 4}{5 \cdot 4} - \frac{1}{2}$

Multiply $\frac{1}{2}$ by $\frac{10}{10}$ .

$\frac{9 \cdot 5}{4 \cdot 5} + \frac{6 \cdot 4}{5 \cdot 4} - (\frac{1}{2} \cdot \frac{10}{10})$

Multiply $\frac{1}{2}$ by $\frac{10}{10}$ .

$\frac{9 \cdot 5}{4 \cdot 5} + \frac{6 \cdot 4}{5 \cdot 4} - \frac{10}{2 \cdot 10}$

Multiply $4$ by $5$ .

$\frac{9 \cdot 5}{20} + \frac{6 \cdot 4}{5 \cdot 4} - \frac{10}{2 \cdot 10}$

Reorder the factors of $5 \cdot 4$ .

$\frac{9 \cdot 5}{20} + \frac{6 \cdot 4}{4 \cdot 5} - \frac{10}{2 \cdot 10}$

Multiply $4$ by $5$ .

$\frac{9 \cdot 5}{20} + \frac{6 \cdot 4}{20} - \frac{10}{2 \cdot 10}$

Multiply $2$ by $10$ .

$\frac{9 \cdot 5}{20} + \frac{6 \cdot 4}{20} - \frac{10}{20}$

Step 3

Combine the numerators over the common denominator.

$\frac{9 \cdot 5 + 6 \cdot 4 - 10}{20}$

Step 4

Simplify each term.

Multiply $9$ by $5$ .

$\frac{45 + 6 \cdot 4 - 10}{20}$

Multiply $6$ by $4$ .

$\frac{45 + 24 - 10}{20}$

Step 5

Simplify by adding and subtracting.

Add $45$ and $24$ .

$\frac{69 - 10}{20}$

Subtract $10$ from $69$ .

$\frac{59}{20}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{59}{20}$

Decimal Form:

$2.95$

Mixed Number Form:

$2 \frac{19}{20}$

Simplify (-2/3)^-2+(3/10)÷(1/4)- square root of (1/2+1/3)÷(10/3)

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