Answer :
To determine which option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], let’s evaluate each choice:
1. Option A: [tex]\(\frac{24}{30}\)[/tex]
Simplifying [tex]\(\frac{24}{30}\)[/tex]:
- Both the numerator and denominator are divisible by 6.
- [tex]\(\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}\)[/tex].
However, [tex]\(\frac{4}{5}\)[/tex] is not the same as [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]. Therefore, this option is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
2. Option B: [tex]\(\frac{4^5}{5}\)[/tex]
This expression simplifies by computing the power of 4 in the numerator and then dividing by 5.
- [tex]\(4^5\)[/tex] is a different value from [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
This means the fraction [tex]\(\frac{4^5}{5}\)[/tex] does not equal [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
3. Option C: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This expression simply means multiplying 6 by [tex]\(\frac{4}{5}\)[/tex].
- [tex]\(6 \cdot \frac{4}{5} = \frac{24}{5}\)[/tex], which is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Therefore, this option is also not correct.
4. Option D: [tex]\(\frac{4^6}{5^6}\)[/tex]
This is the expression of [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]:
- When you raise a fraction to a power, you apply the power to both the numerator and the denominator:
[tex]\(\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}\)[/tex].
So, this option is indeed equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Therefore, the correct answer is Option D: [tex]\(\frac{4^6}{5^6}\)[/tex].
1. Option A: [tex]\(\frac{24}{30}\)[/tex]
Simplifying [tex]\(\frac{24}{30}\)[/tex]:
- Both the numerator and denominator are divisible by 6.
- [tex]\(\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}\)[/tex].
However, [tex]\(\frac{4}{5}\)[/tex] is not the same as [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]. Therefore, this option is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
2. Option B: [tex]\(\frac{4^5}{5}\)[/tex]
This expression simplifies by computing the power of 4 in the numerator and then dividing by 5.
- [tex]\(4^5\)[/tex] is a different value from [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
This means the fraction [tex]\(\frac{4^5}{5}\)[/tex] does not equal [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
3. Option C: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This expression simply means multiplying 6 by [tex]\(\frac{4}{5}\)[/tex].
- [tex]\(6 \cdot \frac{4}{5} = \frac{24}{5}\)[/tex], which is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Therefore, this option is also not correct.
4. Option D: [tex]\(\frac{4^6}{5^6}\)[/tex]
This is the expression of [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]:
- When you raise a fraction to a power, you apply the power to both the numerator and the denominator:
[tex]\(\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}\)[/tex].
So, this option is indeed equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Therefore, the correct answer is Option D: [tex]\(\frac{4^6}{5^6}\)[/tex].
