Sure, let's go through each part of the given question step by step.
### 1. Evaluate the numerical expression [tex]\(\frac{5^{\frac{5}{6}}}{5^{\frac{1}{6}}}\)[/tex]
To simplify this expression, we use the properties of exponents. Specifically, the rule [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]. Here, [tex]\(a = 5\)[/tex], [tex]\(m = \frac{5}{6}\)[/tex], and [tex]\(n = \frac{1}{6}\)[/tex].
[tex]\[
\frac{5^{\frac{5}{6}}}{5^{\frac{1}{6}}} = 5^{\frac{5}{6} - \frac{1}{6}} = 5^{\frac{4}{6}} = 5^{\frac{2}{3}}
\][/tex]
Using this calculation, the value of [tex]\(5^{\frac{2}{3}}\)[/tex] is approximately:
[tex]\[
\frac{5^{\frac{5}{6}}}{5^{\frac{1}{6}}} = 2.924017738212866
\][/tex]
### 2. Calculate [tex]\(\sqrt{15}\)[/tex]
The square root of 15 can be written as [tex]\(15^{\frac{1}{2}}\)[/tex].
The numerical value is:
[tex]\[
\sqrt{15} = 3.872983346207417
\][/tex]
### 3. Calculate [tex]\(\sqrt{125}\)[/tex]
The square root of 125 can be written as [tex]\(125^{\frac{1}{2}}\)[/tex].
The numerical value is:
[tex]\[
\sqrt{125} = 11.180339887498949
\][/tex]
### 4. Calculate [tex]\(\sqrt[3]{10}\)[/tex]
The cube root of 10 can be written as [tex]\(10^{\frac{1}{3}}\)[/tex].
The numerical value is:
[tex]\[
\sqrt[3]{10} = 2.154434690031884
\][/tex]
### 5. Calculate [tex]\(\sqrt[3]{25}\)[/tex]
The cube root of 25 can be written as [tex]\(25^{\frac{1}{3}}\)[/tex].
The numerical value is:
[tex]\[
\sqrt[3]{25} = 2.924017738212866
\][/tex]
To summarize, the evaluated numerical expressions are:
[tex]\[
\frac{5^{\frac{5}{6}}}{5^{\frac{1}{6}}} = 2.924017738212866
\][/tex]
[tex]\[
\sqrt{15} = 3.872983346207417
\][/tex]
[tex]\[
\sqrt{125} = 11.180339887498949
\][/tex]
[tex]\[
\sqrt[3]{10} = 2.154434690031884
\][/tex]
[tex]\[
\sqrt[3]{25} = 2.924017738212866
\][/tex]
