Answer :
To solve the problem, we need to determine which option is equal to the given fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
First, let's understand what [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means:
- [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means we are taking the fraction [tex]\(\frac{4}{5}\)[/tex] and raising it to the power of 6.
- Mathematically, raising a fraction to a power involves raising both the numerator and the denominator to that power.
So, when we calculate [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], we get:
- The numerator becomes [tex]\(4^6\)[/tex].
- The denominator becomes [tex]\(5^6\)[/tex].
Therefore, [tex]\(\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}\)[/tex].
Now, let's match this with the given options:
A. [tex]\(\frac{4^6}{5}\)[/tex] — Here, the denominator is only 5, not [tex]\(5^6\)[/tex].
B. [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex] — This multiplies [tex]\(\frac{4}{5}\)[/tex] by 6, not raising it to the sixth power.
C. [tex]\(\frac{24}{30}\)[/tex] — This simplifies to a fraction which is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
D. [tex]\(\frac{4^6}{5^6}\)[/tex] — This matches exactly with our calculated expression for [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Therefore, the correct answer is D. [tex]\(\frac{4^6}{5^6}\)[/tex].
First, let's understand what [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means:
- [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means we are taking the fraction [tex]\(\frac{4}{5}\)[/tex] and raising it to the power of 6.
- Mathematically, raising a fraction to a power involves raising both the numerator and the denominator to that power.
So, when we calculate [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], we get:
- The numerator becomes [tex]\(4^6\)[/tex].
- The denominator becomes [tex]\(5^6\)[/tex].
Therefore, [tex]\(\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}\)[/tex].
Now, let's match this with the given options:
A. [tex]\(\frac{4^6}{5}\)[/tex] — Here, the denominator is only 5, not [tex]\(5^6\)[/tex].
B. [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex] — This multiplies [tex]\(\frac{4}{5}\)[/tex] by 6, not raising it to the sixth power.
C. [tex]\(\frac{24}{30}\)[/tex] — This simplifies to a fraction which is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
D. [tex]\(\frac{4^6}{5^6}\)[/tex] — This matches exactly with our calculated expression for [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Therefore, the correct answer is D. [tex]\(\frac{4^6}{5^6}\)[/tex].
