Algebra Examples

${(2^{- 2} - 5^{- 1})}^{- 1}$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

${(\frac{1}{4} - 5^{- 1})}^{- 1}$

Step 1.3

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

${(\frac{1}{4} - \frac{1}{5})}^{- 1}$

Step 2

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

${(\frac{1}{4} \cdot \frac{5}{5} - \frac{1}{5})}^{- 1}$

Step 3

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

${(\frac{1}{4} \cdot \frac{5}{5} - \frac{1}{5} \cdot \frac{4}{4})}^{- 1}$

Step 4

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{1}{4}$ by $\frac{5}{5}$ .

${(\frac{5}{4 \cdot 5} - \frac{1}{5} \cdot \frac{4}{4})}^{- 1}$

Step 4.2

Multiply $4$ by $5$ .

${(\frac{5}{20} - \frac{1}{5} \cdot \frac{4}{4})}^{- 1}$

Step 4.3

Multiply $\frac{1}{5}$ by $\frac{4}{4}$ .

${(\frac{5}{20} - \frac{4}{5 \cdot 4})}^{- 1}$

Step 4.4

Multiply $5$ by $4$ .

${(\frac{5}{20} - \frac{4}{20})}^{- 1}$

Step 5

Combine the numerators over the common denominator.

${(\frac{5 - 4}{20})}^{- 1}$

Step 6

Subtract $4$ from $5$ .

${(\frac{1}{20})}^{- 1}$

Step 7

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

$20$