Simplify (2^-1+5^-1)^2

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

Step 1

Simplify each term.

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

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

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

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

Step 2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 3

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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

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

Multiply $2$ by $5$ .

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

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

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

Multiply $5$ by $2$ .

${(\frac{5}{10} + \frac{2}{10})}^{2}$

Step 5

Combine the numerators over the common denominator.

${(\frac{5 + 2}{10})}^{2}$

Step 6

Add $5$ and $2$ .

${(\frac{7}{10})}^{2}$

Step 7

Apply the product rule to $\frac{7}{10}$ .

$\frac{7^{2}}{10^{2}}$

Step 8

Raise $7$ to the power of $2$ .

$\frac{49}{10^{2}}$

Step 9

Raise $10$ to the power of $2$ .

$\frac{49}{100}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{49}{100}$

Decimal Form:

$0.49$