Let's solve the expression step-by-step:
[tex]\[
\frac{25^{\frac{2}{3}} \div 25^{\frac{1}{6}}}{\left(\frac{1}{5}\right)^{\frac{7}{6}} \times \left(\frac{1}{5}\right)^{\frac{1}{6}}}
\][/tex]
### Step 1: Simplify the Numerator
First, let's consider the numerator [tex]\(25^{\frac{2}{3}} \div 25^{\frac{1}{6}}\)[/tex].
Using the property of exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
25^{\frac{2}{3}} \div 25^{\frac{1}{6}} = 25^{\frac{2}{3} - \frac{1}{6}}
\][/tex]
We need to perform the subtraction [tex]\(\frac{2}{3} - \frac{1}{6}\)[/tex]:
Convert [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] to have a common denominator:
[tex]\[
\frac{2}{3} = \frac{4}{6}
\][/tex]
Now, subtract the exponents:
[tex]\[
\frac{4}{6} - \frac{1}{6} = \frac{3}{6} = \frac{1}{2}
\][/tex]
So, the numerator simplifies to:
[tex]\[
25^{\frac{1}{2}}
\][/tex]
### Step 2: Simplify the Denominator
Next, let's simplify the denominator [tex]\(\left(\frac{1}{5}\right)^{\frac{7}{6}} \times \left(\frac{1}{5}\right)^{\frac{1}{6}}\)[/tex].
Using the property of exponents [tex]\(a^m \times a^n = a^{m+n}\)[/tex]:
[tex]\[
\left(\frac{1}{5}\right)^{\frac{7}{6}} \times \left(\frac{1}{5}\right)^{\frac{1}{6}} = \left(\frac{1}{5}\right)^{\frac{7}{6} + \frac{1}{6}}
\][/tex]
Add the exponents:
[tex]\[
\frac{7}{6} + \frac{1}{6} = \frac{8}{6} = \frac{4}{3}
\][/tex]
So, the denominator simplifies to:
[tex]\[
\left(\frac{1}{5}\right)^{\frac{4}{3}}
\][/tex]
### Step 3: Combine the Results
Now, we have:
[tex]\[
\frac{25^{\frac{1}{2}}}{\left(\frac{1}{5}\right)^{\frac{4}{3}}}
\][/tex]
Convert [tex]\(25^{\frac{1}{2}}\)[/tex] and [tex]\(\left(\frac{1}{5}\right)^{\frac{4}{3}}\)[/tex] to more familiar forms:
[tex]\[
25^{\frac{1}{2}} = \sqrt{25} = 5
\][/tex]
[tex]\[
\left(\frac{1}{5}\right)^{\frac{4}{3}} = \left(5^{-1}\right)^{\frac{4}{3}} = 5^{-\frac{4}{3}}
\][/tex]
So, the fraction becomes:
[tex]\[
\frac{5}{5^{-\frac{4}{3}}}
\][/tex]
Using the property of exponents [tex]\(a^m \div a^n = a^{m-n}\)[/tex]:
[tex]\[
5^{1 - \left(-\frac{4}{3}\right)} = 5^{1 + \frac{4}{3}} = 5^{\frac{3}{3} + \frac{4}{3}} = 5^{\frac{7}{3}}
\][/tex]
### Step 4: Final Simplification
The final result is:
[tex]\[
5^{\frac{7}{3}}
\][/tex]
Converting [tex]\(5^{\frac{7}{3}}\)[/tex] to its numerical form:
[tex]\[
5^{\frac{7}{3}} \approx 42.74939866691742
\][/tex]
Thus, the value of the given expression is approximately:
[tex]\[
42.74939866691742
\][/tex]
