Answer :
To determine which of the given options is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], we will evaluate each option individually and compare it to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
First, let's compute [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]:
[tex]\[
\left(\frac{4}{5}\right)^6 \approx 0.262144
\][/tex]
Next, we evaluate each option:
Option A: [tex]\(\frac{4^6}{5}\)[/tex]
[tex]\[
4^6 = 4096 \quad \text{and thus} \quad \frac{4^6}{5} = \frac{4096}{5} = 819.2
\][/tex]
Option B: [tex]\(6 \bullet\left(\frac{4}{5}\right)\)[/tex]
[tex]\[
6 \cdot \frac{4}{5} = 6 \cdot 0.8 = 4.8
\][/tex]
Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
[tex]\[
4^6 = 4096 \quad \text{and} \quad 5^6 = 15625 \quad \text{so} \quad \frac{4^6}{5^6} = \frac{4096}{15625} \approx 0.262144
\][/tex]
Option D: [tex]\(\frac{24}{30}\)[/tex]
[tex]\[
\frac{24}{30} = \frac{4}{5} = 0.8
\][/tex]
Comparing the values, we see:
- [tex]\(\left(\frac{4}{5}\right)^6 \approx 0.262144\)[/tex]
- Option A: [tex]\(\frac{4^6}{5} = 819.2\)[/tex]
- Option B: [tex]\(6 \cdot \left(\frac{4}{5}\right) = 4.8\)[/tex]
- Option C: [tex]\(\frac{4^6}{5^6} \approx 0.262144\)[/tex]
- Option D: [tex]\(\frac{24}{30} = 0.8\)[/tex]
The correct answer is Option C, [tex]\(\frac{4^6}{5^6}\)[/tex], because it equals [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
First, let's compute [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]:
[tex]\[
\left(\frac{4}{5}\right)^6 \approx 0.262144
\][/tex]
Next, we evaluate each option:
Option A: [tex]\(\frac{4^6}{5}\)[/tex]
[tex]\[
4^6 = 4096 \quad \text{and thus} \quad \frac{4^6}{5} = \frac{4096}{5} = 819.2
\][/tex]
Option B: [tex]\(6 \bullet\left(\frac{4}{5}\right)\)[/tex]
[tex]\[
6 \cdot \frac{4}{5} = 6 \cdot 0.8 = 4.8
\][/tex]
Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
[tex]\[
4^6 = 4096 \quad \text{and} \quad 5^6 = 15625 \quad \text{so} \quad \frac{4^6}{5^6} = \frac{4096}{15625} \approx 0.262144
\][/tex]
Option D: [tex]\(\frac{24}{30}\)[/tex]
[tex]\[
\frac{24}{30} = \frac{4}{5} = 0.8
\][/tex]
Comparing the values, we see:
- [tex]\(\left(\frac{4}{5}\right)^6 \approx 0.262144\)[/tex]
- Option A: [tex]\(\frac{4^6}{5} = 819.2\)[/tex]
- Option B: [tex]\(6 \cdot \left(\frac{4}{5}\right) = 4.8\)[/tex]
- Option C: [tex]\(\frac{4^6}{5^6} \approx 0.262144\)[/tex]
- Option D: [tex]\(\frac{24}{30} = 0.8\)[/tex]
The correct answer is Option C, [tex]\(\frac{4^6}{5^6}\)[/tex], because it equals [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
