Answer :
To solve this problem, we need to determine which option is equal to the expression [tex]\((\frac{4}{5})^6\)[/tex].
Let's break it down step-by-step:
1. Understanding the Expression:
The expression [tex]\((\frac{4}{5})^6\)[/tex] means you are raising the fraction [tex]\(\frac{4}{5}\)[/tex] to the 6th power.
2. Applying the Power Rule to Fractions:
When you raise a fraction to a power, you raise both the numerator and the denominator to that power. Therefore:
[tex]\[
\left( \frac{4}{5} \right)^6 = \frac{4^6}{5^6}
\][/tex]
3. Reviewing the Options:
- Option A: [tex]\(\frac{24}{30}\)[/tex] is simply a fraction in its simplest form, which does not involve exponents or powers.
- Option B: [tex]\(\frac{4^6}{5^6}\)[/tex] matches our expression from step 2 exactly.
- Option C: [tex]\(6 \cdot \left( \frac{4}{5} \right)\)[/tex] indicates multiplying the fraction by 6, which is different from raising the fraction to the 6th power.
Based on the breakdown, the expression [tex]\(\frac{4^6}{5^6}\)[/tex] corresponds exactly to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Therefore, the correct answer is Option B: [tex]\(\frac{4^6}{5^6}\)[/tex].
Let's break it down step-by-step:
1. Understanding the Expression:
The expression [tex]\((\frac{4}{5})^6\)[/tex] means you are raising the fraction [tex]\(\frac{4}{5}\)[/tex] to the 6th power.
2. Applying the Power Rule to Fractions:
When you raise a fraction to a power, you raise both the numerator and the denominator to that power. Therefore:
[tex]\[
\left( \frac{4}{5} \right)^6 = \frac{4^6}{5^6}
\][/tex]
3. Reviewing the Options:
- Option A: [tex]\(\frac{24}{30}\)[/tex] is simply a fraction in its simplest form, which does not involve exponents or powers.
- Option B: [tex]\(\frac{4^6}{5^6}\)[/tex] matches our expression from step 2 exactly.
- Option C: [tex]\(6 \cdot \left( \frac{4}{5} \right)\)[/tex] indicates multiplying the fraction by 6, which is different from raising the fraction to the 6th power.
Based on the breakdown, the expression [tex]\(\frac{4^6}{5^6}\)[/tex] corresponds exactly to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Therefore, the correct answer is Option B: [tex]\(\frac{4^6}{5^6}\)[/tex].
