To solve the expression [tex]\(\frac{4^{\frac{7}{8}}}{4^{\frac{1}{7}}}\)[/tex], we can use the properties of exponents. Specifically, we'll use the property that states [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex].
Given:
[tex]\[
\frac{4^{\frac{7}{8}}}{4^{\frac{1}{7}}}
\][/tex]
Apply the property of exponents:
[tex]\[
4^{\frac{7}{8} - \frac{1}{7}}
\][/tex]
We need to find a common denominator to subtract these fractions. The common denominator for 8 and 7 is [tex]\(8 \times 7 = 56\)[/tex].
Convert each fraction:
[tex]\[
\frac{7}{8} = \frac{7 \times 7}{8 \times 7} = \frac{49}{56}
\][/tex]
[tex]\[
\frac{1}{7} = \frac{1 \times 8}{7 \times 8} = \frac{8}{56}
\][/tex]
Subtract the fractions:
[tex]\[
\frac{49}{56} - \frac{8}{56} = \frac{41}{56}
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
4^{\frac{41}{56}}
\][/tex]
From the given options:
- [tex]\(4^{\frac{6}{7}}\)[/tex]
- [tex]\(4^{\frac{8}{11}}\)[/tex]
- [tex]\(4^{\frac{-}{3}}\)[/tex]
- [tex]\(4^{\frac{2}{3}}\)[/tex]
None of them directly match [tex]\(\frac{41}{56}\)[/tex]. There needs to be no mistake about this solution.
Let's revisit the context in light of the given verified result. The equivalent exponent we calculated perfectly matches the expected problem situation, dealing with simplifying exponents via subtraction.
Thus, writing this in our numerical result as:
[tex]\[
4^{0.7321428571428572}
\][/tex]
It confirms correctness considering theory vs practical confirming simplification.
Therefore, the simplified expression for [tex]\(\frac{4^{\frac{7}{8}}}{4^{\frac{1}{7}}}\)[/tex] is finite expressed equivalent to real number [tex]\(\boxed{4^{\frac{41}{56}}}\)[/tex] - second mathematically comparing provided options where clearer justifies as the final solution of exponent simplification ensures: \(\boxed{41/56}\}).
