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Answer :
To determine which option is equal to the fraction [tex]\( \left(\frac{4}{5}\right)^6 \)[/tex], we need to find the value of this expression and compare it with the options:
1. Calculate [tex]\( \left(\frac{4}{5}\right)^6 \)[/tex]:
- This means we want to raise both the numerator (4) and the denominator (5) to the power of 6.
- So, you calculate [tex]\( 4^6 \)[/tex] for the numerator and [tex]\( 5^6 \)[/tex] for the denominator.
2. Compute [tex]\( 4^6 \)[/tex]:
- [tex]\( 4^6 = 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4096 \)[/tex]
3. Compute [tex]\( 5^6 \)[/tex]:
- [tex]\( 5^6 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 15625 \)[/tex]
4. Form the fraction:
- Now, we have [tex]\( \frac{4^6}{5^6} = \frac{4096}{15625} \)[/tex]
5. Compare with the options:
- A. [tex]\(\frac{4^6}{5}\)[/tex]: This is not equivalent as it only raises the numerator to the power 6, not the denominator.
- B. [tex]\(\frac{24}{30}\)[/tex]: Simplifying this gives [tex]\(\frac{4}{5}\)[/tex], not raised to any power.
- C. [tex]\(\frac{4^6}{5^6}\)[/tex]: This matches exactly with [tex]\(\frac{4096}{15625}\)[/tex].
- D. [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex]: This is [tex]\( \frac{24}{5} \)[/tex], different from [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
The correct answer is C. [tex]\(\frac{4^6}{5^6}\)[/tex], which is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
1. Calculate [tex]\( \left(\frac{4}{5}\right)^6 \)[/tex]:
- This means we want to raise both the numerator (4) and the denominator (5) to the power of 6.
- So, you calculate [tex]\( 4^6 \)[/tex] for the numerator and [tex]\( 5^6 \)[/tex] for the denominator.
2. Compute [tex]\( 4^6 \)[/tex]:
- [tex]\( 4^6 = 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4096 \)[/tex]
3. Compute [tex]\( 5^6 \)[/tex]:
- [tex]\( 5^6 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 15625 \)[/tex]
4. Form the fraction:
- Now, we have [tex]\( \frac{4^6}{5^6} = \frac{4096}{15625} \)[/tex]
5. Compare with the options:
- A. [tex]\(\frac{4^6}{5}\)[/tex]: This is not equivalent as it only raises the numerator to the power 6, not the denominator.
- B. [tex]\(\frac{24}{30}\)[/tex]: Simplifying this gives [tex]\(\frac{4}{5}\)[/tex], not raised to any power.
- C. [tex]\(\frac{4^6}{5^6}\)[/tex]: This matches exactly with [tex]\(\frac{4096}{15625}\)[/tex].
- D. [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex]: This is [tex]\( \frac{24}{5} \)[/tex], different from [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
The correct answer is C. [tex]\(\frac{4^6}{5^6}\)[/tex], which is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
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