Let's solve the expression step by step to make sure we understand how to arrive at the final result.
Given the expression:
[tex]\[
\frac{1}{6} + \frac{3 \sqrt{8}}{6} \cdot \left(\frac{4^2}{5}\right) - \sqrt{\frac{9}{85}} \cdot \frac{1}{2}
\][/tex]
We will break it down into parts and simplify each part individually.
Step 1: Simplifying the first term
[tex]\[
\frac{1}{6}
\][/tex]
This term is already in its simplest form:
[tex]\[
\text{Term 1} = \frac{1}{6} \approx 0.1667
\][/tex]
Step 2: Simplifying the second term
First, simplify the fraction:
[tex]\[
\frac{3 \sqrt{8}}{6}
\][/tex]
This can be simplified as follows:
[tex]\[
\frac{3 \sqrt{8}}{6} = \frac{3 \sqrt{8}}{2 \cdot 3} = \frac{\sqrt{8}}{2}
\][/tex]
Now calculate [tex]\(\sqrt{8}\)[/tex]:
[tex]\[
\sqrt{8} = \sqrt{4 \cdot 2} = 2 \sqrt{2}
\][/tex]
So:
[tex]\[
\frac{\sqrt{8}}{2} = \frac{2 \sqrt{2}}{2} = \sqrt{2}
\][/tex]
Next, simplify the rest of the term:
[tex]\[
\sqrt{2} \cdot \left(\frac{4^2}{5}\right)
\][/tex]
Calculate [tex]\(4^2\)[/tex]:
[tex]\[
4^2 = 16
\][/tex]
So the term becomes:
[tex]\[
\sqrt{2} \cdot \left(\frac{16}{5}\right) = \frac{16 \sqrt{2}}{5}
\][/tex]
Thus:
[tex]\[
\text{Term 2} = \frac{3 \sqrt{8}}{6} \cdot \left(\frac{16}{5}\right) = \sqrt{2} \cdot \left(\frac{16}{5}\right) \approx 4.5255
\][/tex]
Step 3: Simplifying the third term
[tex]\[
\sqrt{\frac{9}{85}} \cdot \frac{1}{2}
\][/tex]
First, calculate [tex]\(\sqrt{\frac{9}{85}}\)[/tex]:
[tex]\[
\sqrt{\frac{9}{85}} = \frac{\sqrt{9}}{\sqrt{85}} = \frac{3}{\sqrt{85}}
\][/tex]
We then rationalize the denominator:
[tex]\[
\frac{3}{\sqrt{85}} \cdot \frac{\sqrt{85}}{\sqrt{85}} = \frac{3 \sqrt{85}}{85}
\][/tex]
Therefore:
[tex]\[
\text{Term 3} = \sqrt{\frac{9}{85}} \cdot \frac{1}{2} = \frac{3}{\sqrt{85}} \cdot \frac{1}{2} \approx 0.1627
\][/tex]
Step 4: Combining the terms
Now we add the first two terms and subtract the third term:
[tex]\[
\frac{1}{6} + \frac{3 \sqrt{8}}{6} \cdot \left(\frac{4^2}{5}\right) - \sqrt{\frac{9}{85}} \cdot \frac{1}{2}
\][/tex]
Substituting the simplified values:
[tex]\[
0.1667 + 4.5255 - 0.1627
\][/tex]
Calculating the result:
[tex]\[
0.1667 + 4.5255 = 4.6922
\][/tex]
[tex]\[
4.6922 - 0.1627 = 4.5295
\][/tex]
Thus, the final result of the expression is:
[tex]\[
4.5295
\][/tex]
