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Answer :
To solve the problem of finding which option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], let's take a step-by-step approach:
1. Understand the given expression:
The expression is [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]. This means we need to raise both the numerator (4) and the denominator (5) to the power of 6.
2. Apply the exponent to each part:
- Calculate [tex]\(4^6\)[/tex] for the numerator.
- Calculate [tex]\(5^6\)[/tex] for the denominator.
3. Compute their values:
- [tex]\(4^6 = 4096\)[/tex]
- [tex]\(5^6 = 15625\)[/tex]
4. Form the fraction:
The fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] becomes [tex]\(\frac{4^6}{5^6} = \frac{4096}{15625}\)[/tex].
5. Review the options to find which matches:
- Option A: [tex]\(\frac{24}{30}\)[/tex] is not equal because it simplifies to [tex]\(\frac{4}{5}\)[/tex], not [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option B: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex] simplifies to [tex]\(\frac{24}{5}\)[/tex], which is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option C: [tex]\(\frac{4^6}{5^6}\)[/tex] matches our computation.
- Option D: [tex]\(\frac{4^6}{5}\)[/tex] is obviously not equal due to the denominator not being raised to the 6th power.
Thus, the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is equal to Option C, [tex]\(\frac{4^6}{5^6}\)[/tex].
1. Understand the given expression:
The expression is [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]. This means we need to raise both the numerator (4) and the denominator (5) to the power of 6.
2. Apply the exponent to each part:
- Calculate [tex]\(4^6\)[/tex] for the numerator.
- Calculate [tex]\(5^6\)[/tex] for the denominator.
3. Compute their values:
- [tex]\(4^6 = 4096\)[/tex]
- [tex]\(5^6 = 15625\)[/tex]
4. Form the fraction:
The fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] becomes [tex]\(\frac{4^6}{5^6} = \frac{4096}{15625}\)[/tex].
5. Review the options to find which matches:
- Option A: [tex]\(\frac{24}{30}\)[/tex] is not equal because it simplifies to [tex]\(\frac{4}{5}\)[/tex], not [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option B: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex] simplifies to [tex]\(\frac{24}{5}\)[/tex], which is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option C: [tex]\(\frac{4^6}{5^6}\)[/tex] matches our computation.
- Option D: [tex]\(\frac{4^6}{5}\)[/tex] is obviously not equal due to the denominator not being raised to the 6th power.
Thus, the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is equal to Option C, [tex]\(\frac{4^6}{5^6}\)[/tex].
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