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Answer :
To evaluate [tex]\((125)^{-\frac{2}{3}}\)[/tex], let's break it down step by step:
1. Understand the Negative Exponent: A negative exponent means that we take the reciprocal of the base. So, [tex]\((125)^{-\frac{2}{3}}\)[/tex] is equivalent to [tex]\(\frac{1}{(125)^{\frac{2}{3}}}\)[/tex].
2. Calculate the Fractional Exponent [tex]\(\frac{2}{3}\)[/tex]: The exponent [tex]\(\frac{2}{3}\)[/tex] means we first take the cube root of the number and then square it.
3. Cube Root: Find the cube root of 125. The cube root of 125 is 5 because [tex]\(5 \times 5 \times 5 = 125\)[/tex].
4. Square the Result: Now, take the result from the cube root (which is 5) and square it: [tex]\(5^2 = 25\)[/tex].
5. Reciprocal: Since we had a negative exponent to begin with, take the reciprocal of 25, which is [tex]\(\frac{1}{25}\)[/tex].
Therefore, [tex]\((125)^{-\frac{2}{3}} = \frac{1}{25}\)[/tex].
So, the correct answer is [tex]\(\frac{1}{25}\)[/tex].
1. Understand the Negative Exponent: A negative exponent means that we take the reciprocal of the base. So, [tex]\((125)^{-\frac{2}{3}}\)[/tex] is equivalent to [tex]\(\frac{1}{(125)^{\frac{2}{3}}}\)[/tex].
2. Calculate the Fractional Exponent [tex]\(\frac{2}{3}\)[/tex]: The exponent [tex]\(\frac{2}{3}\)[/tex] means we first take the cube root of the number and then square it.
3. Cube Root: Find the cube root of 125. The cube root of 125 is 5 because [tex]\(5 \times 5 \times 5 = 125\)[/tex].
4. Square the Result: Now, take the result from the cube root (which is 5) and square it: [tex]\(5^2 = 25\)[/tex].
5. Reciprocal: Since we had a negative exponent to begin with, take the reciprocal of 25, which is [tex]\(\frac{1}{25}\)[/tex].
Therefore, [tex]\((125)^{-\frac{2}{3}} = \frac{1}{25}\)[/tex].
So, the correct answer is [tex]\(\frac{1}{25}\)[/tex].
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