Answer :
Let's solve the problem:
We're given the expression [tex]\((\frac{4}{5})^6\)[/tex] and need to find which option among A, B, C, and D is equal to it.
### Step 1: Understand the Expression
The expression [tex]\((\frac{4}{5})^6\)[/tex] means we are raising the fraction [tex]\(\frac{4}{5}\)[/tex] to the sixth power. This involves multiplying [tex]\(\frac{4}{5}\)[/tex] by itself six times:
[tex]\[ \left(\frac{4}{5}\right)^6 = \frac{4}{5} \times \frac{4}{5} \times \frac{4}{5} \times \frac{4}{5} \times \frac{4}{5} \times \frac{4}{5} \][/tex]
### Step 2: Analyze Each Option
- Option A: [tex]\(\frac{4^6}{5}\)[/tex]
- This is not equal to [tex]\((\frac{4}{5})^6\)[/tex] because it only raises 4 to the power of 6 and divides it by 5, instead of raising the entire fraction [tex]\(\frac{4}{5}\)[/tex] to the power of 6.
- Option B: [tex]\(\frac{4^6}{5^6}\)[/tex]
- This is equivalent to [tex]\((\frac{4}{5})^6\)[/tex] because it raises both the numerator (4) and the denominator (5) to the power of 6, which matches the original expression.
- Option C: [tex]\(\frac{24}{30}\)[/tex]
- This fraction reduces to [tex]\(\frac{4}{5}\)[/tex], but it's not raised to the sixth power, so it does not match [tex]\((\frac{4}{5})^6\)[/tex].
- Option D: [tex]\(6 \cdot (\frac{4}{5})\)[/tex]
- This multiplies [tex]\(\frac{4}{5}\)[/tex] by 6, rather than raising it to the power of 6, so it doesn't match [tex]\((\frac{4}{5})^6\)[/tex] either.
### Conclusion
The correct answer is Option B: [tex]\(\frac{4^6}{5^6}\)[/tex], because it correctly represents the entire fraction [tex]\(\frac{4}{5}\)[/tex] raised to the sixth power.
We're given the expression [tex]\((\frac{4}{5})^6\)[/tex] and need to find which option among A, B, C, and D is equal to it.
### Step 1: Understand the Expression
The expression [tex]\((\frac{4}{5})^6\)[/tex] means we are raising the fraction [tex]\(\frac{4}{5}\)[/tex] to the sixth power. This involves multiplying [tex]\(\frac{4}{5}\)[/tex] by itself six times:
[tex]\[ \left(\frac{4}{5}\right)^6 = \frac{4}{5} \times \frac{4}{5} \times \frac{4}{5} \times \frac{4}{5} \times \frac{4}{5} \times \frac{4}{5} \][/tex]
### Step 2: Analyze Each Option
- Option A: [tex]\(\frac{4^6}{5}\)[/tex]
- This is not equal to [tex]\((\frac{4}{5})^6\)[/tex] because it only raises 4 to the power of 6 and divides it by 5, instead of raising the entire fraction [tex]\(\frac{4}{5}\)[/tex] to the power of 6.
- Option B: [tex]\(\frac{4^6}{5^6}\)[/tex]
- This is equivalent to [tex]\((\frac{4}{5})^6\)[/tex] because it raises both the numerator (4) and the denominator (5) to the power of 6, which matches the original expression.
- Option C: [tex]\(\frac{24}{30}\)[/tex]
- This fraction reduces to [tex]\(\frac{4}{5}\)[/tex], but it's not raised to the sixth power, so it does not match [tex]\((\frac{4}{5})^6\)[/tex].
- Option D: [tex]\(6 \cdot (\frac{4}{5})\)[/tex]
- This multiplies [tex]\(\frac{4}{5}\)[/tex] by 6, rather than raising it to the power of 6, so it doesn't match [tex]\((\frac{4}{5})^6\)[/tex] either.
### Conclusion
The correct answer is Option B: [tex]\(\frac{4^6}{5^6}\)[/tex], because it correctly represents the entire fraction [tex]\(\frac{4}{5}\)[/tex] raised to the sixth power.
