Evaluate (2(1/2)^2)/((1/2)^2-1)

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

Step 1

Simplify the numerator.

Rewrite $\frac{1}{2}$ as $2^{- 1}$ .

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

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

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

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

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

Multiply $- 1$ by $1$ .

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

Multiply the exponents in ${(2^{- 1})}^{2}$ .

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

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

Multiply $- 1$ by $2$ .

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

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

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

Subtract $2$ from $1$ .

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

Step 2

Simplify the denominator.

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

$\frac{2^{- 1}}{\frac{1^{2}}{2^{2}} - 1}$

One to any power is one.

$\frac{2^{- 1}}{\frac{1}{2^{2}} - 1}$

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

$\frac{2^{- 1}}{\frac{1}{4} - 1}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{2^{- 1}}{\frac{1}{4} - 1 \cdot \frac{4}{4}}$

Combine $- 1$ and $\frac{4}{4}$ .

$\frac{2^{- 1}}{\frac{1}{4} + \frac{- 1 \cdot 4}{4}}$

Combine the numerators over the common denominator.

$\frac{2^{- 1}}{\frac{1 - 1 \cdot 4}{4}}$

Simplify the numerator.

Multiply $- 1$ by $4$ .

$\frac{2^{- 1}}{\frac{1 - 4}{4}}$

Subtract $4$ from $1$ .

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

Move the negative in front of the fraction.

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

Step 3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{1}{2}}{- \frac{3}{4}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$\frac{1}{2} \cdot \frac{- 4}{3}$

Factor $2$ out of $- 4$ .

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

Cancel the common factor.

$\frac{1}{2} \cdot \frac{2 \cdot - 2}{3}$

Rewrite the expression.

$\frac{- 2}{3}$

Step 6

Move the negative in front of the fraction.

$- \frac{2}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{2}{3}$

Decimal Form:

$- 0.\overline{6}$

Evaluate (2(1/2)^2)/((1/2)^2-1)

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