Of course! Let's evaluate each of the given numerical expressions step-by-step.
### 1. Evaluate [tex]\(\frac{2^{\frac{5}{6}}}{2^{\frac{1}{6}}}\)[/tex]:
Using properties of exponents, specifically the rule [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], we have:
[tex]\[
\frac{2^{\frac{5}{6}}}{2^{\frac{1}{6}}} = 2^{\frac{5}{6} - \frac{1}{6}} = 2^{\frac{4}{6}} = 2^{\frac{2}{3}}
\][/tex]
Evaluating [tex]\(2^{\frac{2}{3}}\)[/tex]:
[tex]\[
2^{\frac{2}{3}} \approx 1.5874
\][/tex]
### 2. Calculate [tex]\(\sqrt[3]{4}\)[/tex]:
The cube root of 4 can be written as [tex]\(4^{\frac{1}{3}}\)[/tex]:
[tex]\[
4^{\frac{1}{3}} \approx 1.5874
\][/tex]
### 3. Calculate [tex]\(\sqrt[3]{2}\)[/tex]:
The cube root of 2 can be written as [tex]\(2^{\frac{1}{3}}\)[/tex]:
[tex]\[
2^{\frac{1}{3}} \approx 1.2599
\][/tex]
### 4. Calculate [tex]\(\sqrt{8}\)[/tex]:
The square root of 8 can be expressed as [tex]\(8^{\frac{1}{2}}\)[/tex]:
[tex]\[
8^{\frac{1}{2}} = \sqrt{8} \approx 2.8284
\][/tex]
### 5. Calculate [tex]\(\sqrt{6}\)[/tex]:
The square root of 6 can be written as [tex]\(6^{\frac{1}{2}}\)[/tex]:
[tex]\[
6^{\frac{1}{2}} = \sqrt{6} \approx 2.4495
\][/tex]
In summary, the evaluated expressions are:
1. [tex]\(\frac{2^{\frac{5}{6}}}{2^{\frac{1}{6}}} \approx 1.5874\)[/tex]
2. [tex]\(\sqrt[3]{4} \approx 1.5874\)[/tex]
3. [tex]\(\sqrt[3]{2} \approx 1.2599\)[/tex]
4. [tex]\(\sqrt{8} \approx 2.8284\)[/tex]
5. [tex]\(\sqrt{6} \approx 2.4495\)[/tex]
These approximations provide the numerical answers to the given expressions.
