Let's break down the given expression step by step: [tex]\(\sqrt{17}(10 \sqrt{17} - 2 \sqrt{17})\)[/tex].
1. Simplify the expression inside the parentheses:
[tex]\[
10 \sqrt{17} - 2 \sqrt{17}
\][/tex]
Since both terms have [tex]\(\sqrt{17}\)[/tex], we can combine them:
[tex]\[
(10 - 2) \sqrt{17} = 8 \sqrt{17}
\][/tex]
So, the expression now is:
[tex]\[
\sqrt{17} \cdot 8 \sqrt{17}
\][/tex]
2. Multiply the terms:
When you multiply [tex]\(\sqrt{17}\)[/tex] by [tex]\(8\sqrt{17}\)[/tex], you can treat [tex]\(\sqrt{17} \cdot \sqrt{17}\)[/tex] as:
[tex]\[
\sqrt{17} \cdot \sqrt{17} = 17
\][/tex]
So, the expression progresses as follows:
[tex]\[
\sqrt{17} \cdot 8 \sqrt{17} = 8 \cdot (\sqrt{17} \cdot \sqrt{17}) = 8 \cdot 17
\][/tex]
3. Perform the multiplication:
[tex]\[
8 \cdot 17 = 136
\][/tex]
Therefore, the simplified value of the given expression [tex]\(\sqrt{17}(10 \sqrt{17} - 2 \sqrt{17})\)[/tex] is [tex]\[ \boxed{136} \][/tex]
Additionally, the intermediate value inside the parentheses is [tex]\(8 \sqrt{17}\)[/tex], which is approximately [tex]\(32.984845004941285\)[/tex].
