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Answer :
Let's find which option is equal to the fraction [tex]\((\frac{4}{5})^6\)[/tex]. We know that:
- The expression [tex]\((\frac{4}{5})^6\)[/tex] represents raising [tex]\(\frac{4}{5}\)[/tex] to the power of 6. This operation will give a result close to 0.262.
Now, let's evaluate each option to compare:
1. Option A: [tex]\(\frac{24}{30}\)[/tex]
- Simplify the fraction: [tex]\(\frac{24}{30} = \frac{4}{5}\)[/tex].
- This equals 0.8, which is much larger than [tex]\((\frac{4}{5})^6\)[/tex].
2. Option B: [tex]\(\frac{4^6}{5}\)[/tex]
- Calculate [tex]\(4^6\)[/tex]: [tex]\(4^6 = 4096\)[/tex].
- This gives: [tex]\(\frac{4096}{5}\)[/tex], which equals 819.2.
- This is much larger than [tex]\((\frac{4}{5})^6\)[/tex].
3. Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
- Calculate both powers: [tex]\(4^6 = 4096\)[/tex] and [tex]\(5^6 = 15625\)[/tex].
- This gives: [tex]\(\frac{4096}{15625}\)[/tex].
- The decimal result is approximately 0.262, which matches [tex]\((\frac{4}{5})^6\)[/tex].
4. Option D: [tex]\(6 \cdot (\binom{4}{5})\)[/tex]
- Evaluate: [tex]\(6 \times \frac{4}{5} = 4.8\)[/tex].
- This is much larger than [tex]\((\frac{4}{5})^6\)[/tex].
By comparing each result with the value of [tex]\((\frac{4}{5})^6\)[/tex], we can see that:
Option C: [tex]\(\frac{4^6}{5^6}\)[/tex] is equal to [tex]\((\frac{4}{5})^6\)[/tex].
- The expression [tex]\((\frac{4}{5})^6\)[/tex] represents raising [tex]\(\frac{4}{5}\)[/tex] to the power of 6. This operation will give a result close to 0.262.
Now, let's evaluate each option to compare:
1. Option A: [tex]\(\frac{24}{30}\)[/tex]
- Simplify the fraction: [tex]\(\frac{24}{30} = \frac{4}{5}\)[/tex].
- This equals 0.8, which is much larger than [tex]\((\frac{4}{5})^6\)[/tex].
2. Option B: [tex]\(\frac{4^6}{5}\)[/tex]
- Calculate [tex]\(4^6\)[/tex]: [tex]\(4^6 = 4096\)[/tex].
- This gives: [tex]\(\frac{4096}{5}\)[/tex], which equals 819.2.
- This is much larger than [tex]\((\frac{4}{5})^6\)[/tex].
3. Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
- Calculate both powers: [tex]\(4^6 = 4096\)[/tex] and [tex]\(5^6 = 15625\)[/tex].
- This gives: [tex]\(\frac{4096}{15625}\)[/tex].
- The decimal result is approximately 0.262, which matches [tex]\((\frac{4}{5})^6\)[/tex].
4. Option D: [tex]\(6 \cdot (\binom{4}{5})\)[/tex]
- Evaluate: [tex]\(6 \times \frac{4}{5} = 4.8\)[/tex].
- This is much larger than [tex]\((\frac{4}{5})^6\)[/tex].
By comparing each result with the value of [tex]\((\frac{4}{5})^6\)[/tex], we can see that:
Option C: [tex]\(\frac{4^6}{5^6}\)[/tex] is equal to [tex]\((\frac{4}{5})^6\)[/tex].
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