Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

$\frac{\frac{1}{4} + 5^{- 1}}{4^{- 1} - 5^{- 1}}$

Step 1.3

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

$\frac{\frac{1}{4} + \frac{1}{5}}{4^{- 1} - 5^{- 1}}$

Step 1.4

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

$\frac{\frac{1}{4} \cdot \frac{5}{5} + \frac{1}{5}}{4^{- 1} - 5^{- 1}}$

Step 1.5

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

$\frac{\frac{1}{4} \cdot \frac{5}{5} + \frac{1}{5} \cdot \frac{4}{4}}{4^{- 1} - 5^{- 1}}$

Step 1.6

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

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Step 1.6.1

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

$\frac{\frac{5}{4 \cdot 5} + \frac{1}{5} \cdot \frac{4}{4}}{4^{- 1} - 5^{- 1}}$

Step 1.6.2

Multiply $4$ by $5$ .

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

Step 1.6.3

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

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

Step 1.6.4

Multiply $5$ by $4$ .

$\frac{\frac{5}{20} + \frac{4}{20}}{4^{- 1} - 5^{- 1}}$

Step 1.7

Combine the numerators over the common denominator.

$\frac{\frac{5 + 4}{20}}{4^{- 1} - 5^{- 1}}$

Step 1.8

Add $5$ and $4$ .

$\frac{\frac{9}{20}}{4^{- 1} - 5^{- 1}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{\frac{9}{20}}{\frac{1}{4} - 5^{- 1}}$

Step 2.2

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

$\frac{\frac{9}{20}}{\frac{1}{4} - \frac{1}{5}}$

Step 2.3

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

$\frac{\frac{9}{20}}{\frac{1}{4} \cdot \frac{5}{5} - \frac{1}{5}}$

Step 2.4

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

$\frac{\frac{9}{20}}{\frac{1}{4} \cdot \frac{5}{5} - \frac{1}{5} \cdot \frac{4}{4}}$

Step 2.5

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

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Step 2.5.1

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

$\frac{\frac{9}{20}}{\frac{5}{4 \cdot 5} - \frac{1}{5} \cdot \frac{4}{4}}$

Step 2.5.2

Multiply $4$ by $5$ .

$\frac{\frac{9}{20}}{\frac{5}{20} - \frac{1}{5} \cdot \frac{4}{4}}$

Step 2.5.3

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

$\frac{\frac{9}{20}}{\frac{5}{20} - \frac{4}{5 \cdot 4}}$

Step 2.5.4

Multiply $5$ by $4$ .

$\frac{\frac{9}{20}}{\frac{5}{20} - \frac{4}{20}}$

Step 2.6

Combine the numerators over the common denominator.

$\frac{\frac{9}{20}}{\frac{5 - 4}{20}}$

Step 2.7

Subtract $4$ from $5$ .

$\frac{\frac{9}{20}}{\frac{1}{20}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{9}{20} \cdot 20$

Step 4

Cancel the common factor of $20$ .

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Step 4.1

Cancel the common factor.

$\frac{9}{20} \cdot 20$

Step 4.2

Rewrite the expression.

$9$