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Answer :
To solve the expression [tex]\((-125)^{2/3}\)[/tex], we can break it down into steps:
1. Understand the expression [tex]\((-125)^{2/3}\)[/tex]:
The expression involves a fractional exponent, which means it can be rewritten in terms of radicals and powers. We interpret [tex]\(a^{p/q}\)[/tex] as [tex]\(\sqrt[q]{a^p}\)[/tex].
2. Breaking down the exponent [tex]\((2/3)\)[/tex]:
[tex]\((-125)^{2/3}\)[/tex] can be rewritten as [tex]\(((-125)^{1/3})^2\)[/tex], meaning we first find the cube root of -125, and then square the result.
3. Finding the cube root of -125:
The cube root of -125 is the number which, when multiplied by itself twice, gives -125. Since [tex]\((-5) \times (-5) \times (-5) = -125\)[/tex], the cube root of -125 is [tex]\(-5\)[/tex].
4. Squaring the cube root:
Now we need to square the result from the previous step. So, [tex]\((-5)^2 = 25\)[/tex].
Therefore, the value of the expression [tex]\((-125)^{2/3}\)[/tex] is [tex]\(25\)[/tex].
So, the correct answer in the given choices is [tex]\(e) 25\)[/tex].
1. Understand the expression [tex]\((-125)^{2/3}\)[/tex]:
The expression involves a fractional exponent, which means it can be rewritten in terms of radicals and powers. We interpret [tex]\(a^{p/q}\)[/tex] as [tex]\(\sqrt[q]{a^p}\)[/tex].
2. Breaking down the exponent [tex]\((2/3)\)[/tex]:
[tex]\((-125)^{2/3}\)[/tex] can be rewritten as [tex]\(((-125)^{1/3})^2\)[/tex], meaning we first find the cube root of -125, and then square the result.
3. Finding the cube root of -125:
The cube root of -125 is the number which, when multiplied by itself twice, gives -125. Since [tex]\((-5) \times (-5) \times (-5) = -125\)[/tex], the cube root of -125 is [tex]\(-5\)[/tex].
4. Squaring the cube root:
Now we need to square the result from the previous step. So, [tex]\((-5)^2 = 25\)[/tex].
Therefore, the value of the expression [tex]\((-125)^{2/3}\)[/tex] is [tex]\(25\)[/tex].
So, the correct answer in the given choices is [tex]\(e) 25\)[/tex].
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