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Answer :
Let's solve the expression [tex]\(\left((-625)^2\right)^{-\frac{1}{8}}\)[/tex] step by step.
1. Calculate [tex]\((-625)^2\)[/tex]:
- When we square [tex]\(-625\)[/tex], we get [tex]\( (-625) \times (-625) \)[/tex].
- This results in a positive number because the product of two negative numbers is positive.
- [tex]\((-625)^2 = 390625\)[/tex].
2. Now, raise 390625 to the power of [tex]\(-\frac{1}{8}\)[/tex]:
- Raising a number to a negative exponent means taking the reciprocal of the number raised to the positive exponent.
- So, [tex]\((390625)^{-\frac{1}{8}} = \frac{1}{(390625)^{\frac{1}{8}}}\)[/tex].
- Calculate [tex]\((390625)^{\frac{1}{8}}\)[/tex]. This is asking for the eighth root of 390625.
3. Find the eighth root of 390625:
- The eighth root of 390625 is 5 because [tex]\(5^8 = 390625\)[/tex].
4. Take the reciprocal:
- Since we wanted [tex]\((390625)^{-\frac{1}{8}}\)[/tex], take the reciprocal of 5.
- The reciprocal of 5 is [tex]\(\frac{1}{5}\)[/tex].
Therefore, the result of the expression is [tex]\(\frac{1}{5}\)[/tex].
So, the correct answer is option D) [tex]\(\frac{1}{5}\)[/tex].
1. Calculate [tex]\((-625)^2\)[/tex]:
- When we square [tex]\(-625\)[/tex], we get [tex]\( (-625) \times (-625) \)[/tex].
- This results in a positive number because the product of two negative numbers is positive.
- [tex]\((-625)^2 = 390625\)[/tex].
2. Now, raise 390625 to the power of [tex]\(-\frac{1}{8}\)[/tex]:
- Raising a number to a negative exponent means taking the reciprocal of the number raised to the positive exponent.
- So, [tex]\((390625)^{-\frac{1}{8}} = \frac{1}{(390625)^{\frac{1}{8}}}\)[/tex].
- Calculate [tex]\((390625)^{\frac{1}{8}}\)[/tex]. This is asking for the eighth root of 390625.
3. Find the eighth root of 390625:
- The eighth root of 390625 is 5 because [tex]\(5^8 = 390625\)[/tex].
4. Take the reciprocal:
- Since we wanted [tex]\((390625)^{-\frac{1}{8}}\)[/tex], take the reciprocal of 5.
- The reciprocal of 5 is [tex]\(\frac{1}{5}\)[/tex].
Therefore, the result of the expression is [tex]\(\frac{1}{5}\)[/tex].
So, the correct answer is option D) [tex]\(\frac{1}{5}\)[/tex].
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