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Answer
The simplified expression is \( (a+b)^{-2} \).
Solution
To simplify the expression, start by rewriting the cube root as an exponent:
√[3]{(a+b)^(–7)} = (a+b)^(–7/3).
Now, multiply by (a+b)^(1/3):
(a+b)^(–7/3) × (a+b)^(1/3).
When multiplying expressions with the same base, add the exponents:
–7/3 + 1/3 = (–7 + 1)/3 = –6/3 = –2.
So, the simplified expression is:
(a+b)^(–2).
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Simplify this solution Beyond the Answer
To simplify the expression \( \sqrt[3]{(a+b)^{-7}} \times (a+b)^{\frac{1}{3}} \), we can rewrite the cube root as an exponent: \[ \sqrt[3]{(a+b)^{-7}} = (a+b)^{-\frac{7}{3}}. \] Now, substituting back into the expression gives: \[ (a+b)^{-\frac{7}{3}} \times (a+b)^{\frac{1}{3}}. \] When multiplying expressions with the same base, we add the exponents: \[ = (a+b)^{-\frac{7}{3} + \frac{1}{3}} = (a+b)^{-\frac{6}{3}} = (a+b)^{-2}. \] Therefore, the simplified expression is: \[ \frac{1}{(a+b)^{2}}. \]
