Let's expand and simplify the given expression: [tex]\(\sqrt{10}(\sqrt{10} - \sqrt{2}) + 8 \sqrt{5} \)[/tex].
### Step 1: Expand the expression [tex]\(\sqrt{10}(\sqrt{10} - \sqrt{2})\)[/tex]
First, distribute [tex]\(\sqrt{10}\)[/tex] across the terms inside the parenthesis:
[tex]\[
\sqrt{10} \cdot \sqrt{10} - \sqrt{10} \cdot \sqrt{2}
\][/tex]
### Step 2: Simplify each term
- The product of [tex]\(\sqrt{10} \cdot \sqrt{10}\)[/tex]:
[tex]\[
\sqrt{10} \cdot \sqrt{10} = 10
\][/tex]
- The product of [tex]\(\sqrt{10} \cdot -\sqrt{2}\)[/tex]:
[tex]\[
\sqrt{10} \cdot -\sqrt{2} = -\sqrt{20}
\][/tex]
### Step 3: Combine the results from Step 2
Putting these results together, we have:
[tex]\[
10 - \sqrt{20}
\][/tex]
### Step 4: Simplify [tex]\(-\sqrt{20}\)[/tex]
We recognize that:
[tex]\[
\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \cdot \sqrt{5} = 2 \sqrt{5}
\][/tex]
Thus:
[tex]\[
-\sqrt{20} = -2 \sqrt{5}
\][/tex]
Hence, the expression simplifies to:
[tex]\[
10 - 2 \sqrt{5}
\][/tex]
### Step 5: Add [tex]\(8 \sqrt{5}\)[/tex] to the simplified expression
Now add [tex]\(8 \sqrt{5}\)[/tex] to [tex]\(10 - 2 \sqrt{5}\)[/tex]:
[tex]\[
10 - 2 \sqrt{5} + 8 \sqrt{5}
\][/tex]
Combine the like terms involving [tex]\(\sqrt{5}\)[/tex]:
[tex]\[
10 + (8 \sqrt{5} - 2 \sqrt{5}) = 10 + 6 \sqrt{5}
\][/tex]
### Step 6: Final simplified form
The final simplified form of the expression [tex]\(\sqrt{10}(\sqrt{10} - \sqrt{2}) + 8 \sqrt{5}\)[/tex] is:
[tex]\[
10 + 6 \sqrt{5}
\][/tex]
If we convert this to a numerical value for confirmation, we get:
- Calculate [tex]\(6 \sqrt{5}\)[/tex]:
[tex]\[
6 \sqrt{5} \approx 6 \times 2.236 = 13.416
\][/tex]
- Therefore, the final numerical result is:
[tex]\[
10 + 13.416 = 23.416
\][/tex]
The final answer is:
[tex]\[
23.416
\][/tex]
