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Chapter 6: Chapter 6

Problem 70

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

Short Answer

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The simplified expression is \(\frac{36}{25}\).
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TABLE OF CONTENTS :
TABLE OF CONTENTS

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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