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Problem 70

# Simplify the expression $$\left(3^{-1}+2^{-1}\right)^{-2}$$

### Short Answer

Expert verified
The simplified expression is $$\frac{36}{25}$$.
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## Step 1: 1. Rewrite the expression as fractions

Given the expression $\left(3^{-1}+2^{-1}\right)^{-2},$ first rewrite the negative exponents as fractions: $\left(\frac{1}{3}+\frac{1}{2}\right)^{-2}.$

## Step 2: 2. Find a common denominator

To add the fractions inside the parenthesis, we need to find a common denominator. Since the denominators are 3 and 2, the least common multiple is 6. We will rewrite each fraction with the common denominator: $\left(\frac{2}{6}+\frac{3}{6}\right)^{-2}.$

## Step 3: 3. Add the fractions

Now that both fractions have a common denominator, we can add them: $\left(\frac{2}{6}+\frac{3}{6}\right)^{-2} = \left(\frac{5}{6}\right)^{-2}.$

## Step 4: 4. Simplify the expression using exponent properties

Since the exponent outside the parenthesis is -2, we can rewrite the expression as a fraction, inverting the base and squaring it: $\left(\frac{5}{6}\right)^{-2} = \left(\frac{6}{5}\right)^{2}.$ Now, we just need to raise $$\frac{6}{5}$$ to the power of $$2$$: $\left(\frac{6}{5}\right)^{2} = \frac{6^2}{5^2} = \frac{36}{25}.$ The simplified expression is $$\boxed{\frac{36}{25}}$$.

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