Answer :
To determine which of the given options is equal to the fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], let's evaluate each choice and compare it to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
1. Original Expression:
[tex]\[
\left(\frac{4}{5}\right)^6
\][/tex]
This expression calculates the fraction [tex]\(\frac{4}{5}\)[/tex] raised to the 6th power.
2. Option A: [tex]\(\frac{24}{30}\)[/tex]
Simplifying [tex]\(\frac{24}{30}\)[/tex]:
- The greatest common divisor (GCD) of 24 and 30 is 6.
- Simplifying gives us [tex]\(\frac{24 \div 6}{30 \div 6} = \frac{4}{5}\)[/tex].
The simplified fraction is [tex]\(\frac{4}{5}\)[/tex], not [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
3. Option B: [tex]\(\frac{4^6}{5}\)[/tex]
This is simply [tex]\(4^6\)[/tex] divided by 5, not [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
4. Option C: [tex]\(6 \times \left(\frac{4}{5}\right)\)[/tex]
This means multiplying [tex]\(\frac{4}{5}\)[/tex] by 6, giving [tex]\(\frac{24}{5}\)[/tex], which is not [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
5. Option D: [tex]\(\frac{4^6}{5^6}\)[/tex]
This is exactly the expression of [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], because:
- [tex]\(\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}\)[/tex] according to the properties of exponents.
Therefore, option D, [tex]\(\frac{4^6}{5^6}\)[/tex], is the correct expression equivalent to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
1. Original Expression:
[tex]\[
\left(\frac{4}{5}\right)^6
\][/tex]
This expression calculates the fraction [tex]\(\frac{4}{5}\)[/tex] raised to the 6th power.
2. Option A: [tex]\(\frac{24}{30}\)[/tex]
Simplifying [tex]\(\frac{24}{30}\)[/tex]:
- The greatest common divisor (GCD) of 24 and 30 is 6.
- Simplifying gives us [tex]\(\frac{24 \div 6}{30 \div 6} = \frac{4}{5}\)[/tex].
The simplified fraction is [tex]\(\frac{4}{5}\)[/tex], not [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
3. Option B: [tex]\(\frac{4^6}{5}\)[/tex]
This is simply [tex]\(4^6\)[/tex] divided by 5, not [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
4. Option C: [tex]\(6 \times \left(\frac{4}{5}\right)\)[/tex]
This means multiplying [tex]\(\frac{4}{5}\)[/tex] by 6, giving [tex]\(\frac{24}{5}\)[/tex], which is not [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
5. Option D: [tex]\(\frac{4^6}{5^6}\)[/tex]
This is exactly the expression of [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], because:
- [tex]\(\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}\)[/tex] according to the properties of exponents.
Therefore, option D, [tex]\(\frac{4^6}{5^6}\)[/tex], is the correct expression equivalent to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
