Evaluate ((1-3/10)÷((2/3+1/2)/(1/2*(4/5-1))))^-2

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify the numerator.

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Combine $- 1$ and $\frac{5}{5}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $5$ .

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

Subtract $5$ from $4$ .

${((1 - \frac{3}{10}) \frac{\frac{1}{2} \cdot \frac{- 1}{5}}{\frac{2}{3} + \frac{1}{2}})}^{- 2}$

Move the negative in front of the fraction.

${((1 - \frac{3}{10}) \frac{\frac{1}{2} \cdot - 1 \frac{1}{5}}{\frac{2}{3} + \frac{1}{2}})}^{- 2}$

Combine exponents.

Factor out negative.

${((1 - \frac{3}{10}) \frac{- (\frac{1}{2} \cdot \frac{1}{5})}{\frac{2}{3} + \frac{1}{2}})}^{- 2}$

Multiply $\frac{1}{2}$ by $\frac{1}{5}$ .

${((1 - \frac{3}{10}) \frac{- \frac{1}{2 \cdot 5}}{\frac{2}{3} + \frac{1}{2}})}^{- 2}$

Multiply $2$ by $5$ .

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

Step 3

Simplify the denominator.

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

${((1 - \frac{3}{10}) \frac{- \frac{1}{10}}{\frac{2}{3} \cdot \frac{2}{2} + \frac{1}{2}})}^{- 2}$

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

${((1 - \frac{3}{10}) \frac{- \frac{1}{10}}{\frac{2}{3} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{3}{3}})}^{- 2}$

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

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

${((1 - \frac{3}{10}) \frac{- \frac{1}{10}}{\frac{2 \cdot 2}{3 \cdot 2} + \frac{1}{2} \cdot \frac{3}{3}})}^{- 2}$

Multiply $3$ by $2$ .

${((1 - \frac{3}{10}) \frac{- \frac{1}{10}}{\frac{2 \cdot 2}{6} + \frac{1}{2} \cdot \frac{3}{3}})}^{- 2}$

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

${((1 - \frac{3}{10}) \frac{- \frac{1}{10}}{\frac{2 \cdot 2}{6} + \frac{3}{2 \cdot 3}})}^{- 2}$

Multiply $2$ by $3$ .

${((1 - \frac{3}{10}) \frac{- \frac{1}{10}}{\frac{2 \cdot 2}{6} + \frac{3}{6}})}^{- 2}$

Combine the numerators over the common denominator.

${((1 - \frac{3}{10}) \frac{- \frac{1}{10}}{\frac{2 \cdot 2 + 3}{6}})}^{- 2}$

Simplify the numerator.

Multiply $2$ by $2$ .

${((1 - \frac{3}{10}) \frac{- \frac{1}{10}}{\frac{4 + 3}{6}})}^{- 2}$

Add $4$ and $3$ .

${((1 - \frac{3}{10}) \frac{- \frac{1}{10}}{\frac{7}{6}})}^{- 2}$

Step 4

Simplify the expression.

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

${((\frac{10}{10} - \frac{3}{10}) \frac{- \frac{1}{10}}{\frac{7}{6}})}^{- 2}$

Combine the numerators over the common denominator.

${(\frac{10 - 3}{10} \cdot \frac{- \frac{1}{10}}{\frac{7}{6}})}^{- 2}$

Subtract $3$ from $10$ .

${(\frac{7}{10} \cdot \frac{- \frac{1}{10}}{\frac{7}{6}})}^{- 2}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

${(\frac{7}{10} (- \frac{1}{10} \cdot \frac{6}{7}))}^{- 2}$

Step 6

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{10}$ into the numerator.

${(\frac{7}{10} (\frac{- 1}{10} \cdot \frac{6}{7}))}^{- 2}$

Factor $2$ out of $10$ .

${(\frac{7}{10} (\frac{- 1}{2 (5)} \cdot \frac{6}{7}))}^{- 2}$

Factor $2$ out of $6$ .

${(\frac{7}{10} (\frac{- 1}{2 \cdot 5} \cdot \frac{2 \cdot 3}{7}))}^{- 2}$

Cancel the common factor.

${(\frac{7}{10} (\frac{- 1}{2 \cdot 5} \cdot \frac{2 \cdot 3}{7}))}^{- 2}$

Rewrite the expression.

${(\frac{7}{10} (\frac{- 1}{5} \cdot \frac{3}{7}))}^{- 2}$

Step 7

Multiply $\frac{- 1}{5}$ by $\frac{3}{7}$ .

${(\frac{7}{10} \cdot \frac{- 1 \cdot 3}{5 \cdot 7})}^{- 2}$

Step 8

Multiply.

Multiply $- 1$ by $3$ .

${(\frac{7}{10} \cdot \frac{- 3}{5 \cdot 7})}^{- 2}$

Multiply $5$ by $7$ .

${(\frac{7}{10} \cdot \frac{- 3}{35})}^{- 2}$

Step 9

Cancel the common factor of $7$ .

Factor $7$ out of $35$ .

${(\frac{7}{10} \cdot \frac{- 3}{7 (5)})}^{- 2}$

Cancel the common factor.

${(\frac{7}{10} \cdot \frac{- 3}{7 \cdot 5})}^{- 2}$

Rewrite the expression.

${(\frac{1}{10} \cdot \frac{- 3}{5})}^{- 2}$

Step 10

Multiply $\frac{1}{10}$ by $\frac{- 3}{5}$ .

${(\frac{- 3}{10 \cdot 5})}^{- 2}$

Step 11

Simplify the expression.

Multiply $10$ by $5$ .

${(\frac{- 3}{50})}^{- 2}$

Move the negative in front of the fraction.

${(- \frac{3}{50})}^{- 2}$

Step 12

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

${(- \frac{50}{3})}^{2}$

Step 13

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

Apply the product rule to $- \frac{50}{3}$ .

${(- 1)}^{2} {(\frac{50}{3})}^{2}$

Apply the product rule to $\frac{50}{3}$ .

${(- 1)}^{2} \frac{50^{2}}{3^{2}}$

Step 14

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

$1 \frac{50^{2}}{3^{2}}$

Step 15

Multiply $\frac{50^{2}}{3^{2}}$ by $1$ .

$\frac{50^{2}}{3^{2}}$

Step 16

Raise $50$ to the power of $2$ .

$\frac{2500}{3^{2}}$

Step 17

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

$\frac{2500}{9}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$\frac{2500}{9}$

Decimal Form:

$277.\overline{7}$

Mixed Number Form:

$277 \frac{7}{9}$

Evaluate ((1-3/10)÷((2/3+1/2)/(1/2*(4/5-1))))^-2

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