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Answer :
To find which option is equal to [tex]\((\frac{4}{5})^6\)[/tex], we need to evaluate this expression.
1. Calculate [tex]\((4/5)^6\)[/tex]:
- [tex]\((4/5)^6\)[/tex] means raising both the numerator and the denominator to the power of 6.
- Calculate [tex]\(4^6\)[/tex]: When you multiply 4 by itself 6 times, you get 4096.
- Calculate [tex]\(5^6\)[/tex]: When you multiply 5 by itself 6 times, you get 15625.
- So, [tex]\((4/5)^6 = \frac{4096}{15625}\)[/tex].
Now, let's compare this result with the options:
- Option A: [tex]\(6 \cdot (\frac{4}{5})\)[/tex]
- This is not the same as [tex]\((\frac{4}{5})^6\)[/tex] because you're multiplying [tex]\(\frac{4}{5}\)[/tex] by 6, not raising it to the 6th power.
- Option B: [tex]\(\frac{4^6}{5}\)[/tex]
- This represents [tex]\(\frac{4096}{5}\)[/tex], which is not the same as [tex]\(\frac{4096}{15625}\)[/tex].
- Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
- This is [tex]\(\frac{4096}{15625}\)[/tex], which matches the calculated expression [tex]\((\frac{4}{5})^6\)[/tex].
- Option D: [tex]\(\frac{24}{30}\)[/tex]
- This simplifies to [tex]\(\frac{4}{5}\)[/tex], not [tex]\((\frac{4}{5})^6\)[/tex].
The correct option is C, [tex]\(\frac{4^6}{5^6}\)[/tex], which is equivalent to [tex]\((\frac{4}{5})^6\)[/tex].
1. Calculate [tex]\((4/5)^6\)[/tex]:
- [tex]\((4/5)^6\)[/tex] means raising both the numerator and the denominator to the power of 6.
- Calculate [tex]\(4^6\)[/tex]: When you multiply 4 by itself 6 times, you get 4096.
- Calculate [tex]\(5^6\)[/tex]: When you multiply 5 by itself 6 times, you get 15625.
- So, [tex]\((4/5)^6 = \frac{4096}{15625}\)[/tex].
Now, let's compare this result with the options:
- Option A: [tex]\(6 \cdot (\frac{4}{5})\)[/tex]
- This is not the same as [tex]\((\frac{4}{5})^6\)[/tex] because you're multiplying [tex]\(\frac{4}{5}\)[/tex] by 6, not raising it to the 6th power.
- Option B: [tex]\(\frac{4^6}{5}\)[/tex]
- This represents [tex]\(\frac{4096}{5}\)[/tex], which is not the same as [tex]\(\frac{4096}{15625}\)[/tex].
- Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
- This is [tex]\(\frac{4096}{15625}\)[/tex], which matches the calculated expression [tex]\((\frac{4}{5})^6\)[/tex].
- Option D: [tex]\(\frac{24}{30}\)[/tex]
- This simplifies to [tex]\(\frac{4}{5}\)[/tex], not [tex]\((\frac{4}{5})^6\)[/tex].
The correct option is C, [tex]\(\frac{4^6}{5^6}\)[/tex], which is equivalent to [tex]\((\frac{4}{5})^6\)[/tex].
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