Answer :
To solve the problem, we need to determine which option is equal to the fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
### Step-by-Step Solution:
1. Understanding the Expression:
The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means that both the numerator and the denominator of the fraction [tex]\(\frac{4}{5}\)[/tex] are raised to the power of 6.
2. Applying the Power Rule:
When we raise a fraction to a power, we apply the power to both the numerator and the denominator separately. Therefore:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}
\][/tex]
3. Matching the Options:
- Option A: [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex] is [tex]\(\frac{24}{5}\)[/tex], which is not the same as [tex]\(\frac{4^6}{5^6}\)[/tex].
- Option B: [tex]\(\frac{4^6}{5^6}\)[/tex] directly matches our expression.
- Option C: [tex]\(\frac{4^6}{5}\)[/tex] only the numerator is raised to the power of 6, which is incorrect.
- Option D: [tex]\(\frac{24}{30}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex], which does not involve raising to any power.
Since [tex]\(\frac{4^6}{5^6}\)[/tex] matches our derived expression, the correct answer is Option B: [tex]\(\frac{4^6}{5^6}\)[/tex].
### Step-by-Step Solution:
1. Understanding the Expression:
The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means that both the numerator and the denominator of the fraction [tex]\(\frac{4}{5}\)[/tex] are raised to the power of 6.
2. Applying the Power Rule:
When we raise a fraction to a power, we apply the power to both the numerator and the denominator separately. Therefore:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}
\][/tex]
3. Matching the Options:
- Option A: [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex] is [tex]\(\frac{24}{5}\)[/tex], which is not the same as [tex]\(\frac{4^6}{5^6}\)[/tex].
- Option B: [tex]\(\frac{4^6}{5^6}\)[/tex] directly matches our expression.
- Option C: [tex]\(\frac{4^6}{5}\)[/tex] only the numerator is raised to the power of 6, which is incorrect.
- Option D: [tex]\(\frac{24}{30}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex], which does not involve raising to any power.
Since [tex]\(\frac{4^6}{5^6}\)[/tex] matches our derived expression, the correct answer is Option B: [tex]\(\frac{4^6}{5^6}\)[/tex].
