Answer :
To solve the problem of determining which option is equal to the fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], let's examine each choice step-by-step.
### Step 1: Calculate the Fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]
The given fraction is [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
### Step 2: Simplify Each Option
Now, let's simplify and compare each given option:
#### Option A: [tex]\(\frac{24}{30}\)[/tex]
Simplify [tex]\(\frac{24}{30}\)[/tex]:
[tex]\[\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}\][/tex]
This option represents [tex]\(\frac{4}{5}\)[/tex], not [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], so it is not correct.
#### Option B: [tex]\(\frac{4^6}{5^6}\)[/tex]
Calculate [tex]\(\frac{4^6}{5^6}\)[/tex]:
[tex]\[
\frac{4^6}{5^6} = \left(\frac{4}{5}\right)^6
\][/tex]
This matches exactly with our fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
#### Option C: [tex]\(\frac{4^6}{5}\)[/tex]
To check if this equals [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], compute:
[tex]\[
\frac{4^6}{5} \neq \left(\frac{4}{5}\right)^6
\][/tex]
So, this is not correct.
#### Option D: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
Calculate this expression:
[tex]\[
6 \cdot \left(\frac{4}{5}\right) = \frac{24}{5}
\][/tex]
This does not equal [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], so this option is also not correct.
### Conclusion
Only Option B: [tex]\(\frac{4^6}{5^6}\)[/tex] simplifies to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]. Thus, Option B is the correct answer.
### Step 1: Calculate the Fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]
The given fraction is [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
### Step 2: Simplify Each Option
Now, let's simplify and compare each given option:
#### Option A: [tex]\(\frac{24}{30}\)[/tex]
Simplify [tex]\(\frac{24}{30}\)[/tex]:
[tex]\[\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}\][/tex]
This option represents [tex]\(\frac{4}{5}\)[/tex], not [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], so it is not correct.
#### Option B: [tex]\(\frac{4^6}{5^6}\)[/tex]
Calculate [tex]\(\frac{4^6}{5^6}\)[/tex]:
[tex]\[
\frac{4^6}{5^6} = \left(\frac{4}{5}\right)^6
\][/tex]
This matches exactly with our fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
#### Option C: [tex]\(\frac{4^6}{5}\)[/tex]
To check if this equals [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], compute:
[tex]\[
\frac{4^6}{5} \neq \left(\frac{4}{5}\right)^6
\][/tex]
So, this is not correct.
#### Option D: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
Calculate this expression:
[tex]\[
6 \cdot \left(\frac{4}{5}\right) = \frac{24}{5}
\][/tex]
This does not equal [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], so this option is also not correct.
### Conclusion
Only Option B: [tex]\(\frac{4^6}{5^6}\)[/tex] simplifies to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]. Thus, Option B is the correct answer.
