Evaluate (3^-1+2^-2)/(3^-1+2^2)

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

Step 1

Simplify the numerator.

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

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

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

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

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

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

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

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

Multiply $3$ by $4$ .

$\frac{\frac{4}{12} + \frac{1}{4} \cdot \frac{3}{3}}{3^{- 1} + 2^{2}}$

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

$\frac{\frac{4}{12} + \frac{3}{4 \cdot 3}}{3^{- 1} + 2^{2}}$

Multiply $4$ by $3$ .

$\frac{\frac{4}{12} + \frac{3}{12}}{3^{- 1} + 2^{2}}$

Combine the numerators over the common denominator.

$\frac{\frac{4 + 3}{12}}{3^{- 1} + 2^{2}}$

Add $4$ and $3$ .

$\frac{\frac{7}{12}}{3^{- 1} + 2^{2}}$

Step 2

Simplify the denominator.

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

$\frac{\frac{7}{12}}{\frac{1}{3} + 2^{2}}$

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

$\frac{\frac{7}{12}}{\frac{1}{3} + 4}$

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{7}{12}}{\frac{1}{3} + 4 \cdot \frac{3}{3}}$

Combine $4$ and $\frac{3}{3}$ .

$\frac{\frac{7}{12}}{\frac{1}{3} + \frac{4 \cdot 3}{3}}$

Combine the numerators over the common denominator.

$\frac{\frac{7}{12}}{\frac{1 + 4 \cdot 3}{3}}$

Simplify the numerator.

Multiply $4$ by $3$ .

$\frac{\frac{7}{12}}{\frac{1 + 12}{3}}$

Add $1$ and $12$ .

$\frac{\frac{7}{12}}{\frac{13}{3}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{7}{12} \cdot \frac{3}{13}$

Step 4

Cancel the common factor of $3$ .

Factor $3$ out of $12$ .

$\frac{7}{3 (4)} \cdot \frac{3}{13}$

Cancel the common factor.

$\frac{7}{3 \cdot 4} \cdot \frac{3}{13}$

Rewrite the expression.

$\frac{7}{4} \cdot \frac{1}{13}$

Step 5

Multiply $\frac{7}{4}$ by $\frac{1}{13}$ .

$\frac{7}{4 \cdot 13}$

Step 6

Multiply $4$ by $13$ .

$\frac{7}{52}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{52}$

Decimal Form:

$0.13461538 \dots$

Evaluate (3^-1+2^-2)/(3^-1+2^2)

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