Certainly! Let's solve the given mathematical expression step-by-step.
Given expression:
[tex]\[ \sqrt{5}(6 - 5\sqrt{2}) \][/tex]
We're asked to simplify this expression.
### Step-by-Step Solution:
1. Identify each term:
- The expression involves the product of [tex]\( \sqrt{5} \)[/tex] and the binomial [tex]\( (6 - 5\sqrt{2}) \)[/tex].
2. Distribute [tex]\( \sqrt{5} \)[/tex] over the binomial:
[tex]\[
\sqrt{5}(6 - 5\sqrt{2})
\][/tex]
3. Apply the distributive property:
Using the distributive property (also known as the distributive law), we multiply [tex]\( \sqrt{5} \)[/tex] by each term inside the parentheses:
[tex]\[
\sqrt{5} \cdot 6 - \sqrt{5} \cdot 5\sqrt{2}
\][/tex]
Let's break it down:
- First term: [tex]\( \sqrt{5} \cdot 6 \)[/tex]
[tex]\[
6\sqrt{5}
\][/tex]
- Second term: [tex]\( \sqrt{5} \cdot 5\sqrt{2} \)[/tex]
[tex]\[
5\sqrt{5} \cdot \sqrt{2}
\][/tex]
4. Combine the second term's radicals:
Recall that [tex]\( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \)[/tex]:
[tex]\[
5\sqrt{5 \cdot 2}
\][/tex]
Inside the radical, multiply 5 and 2:
[tex]\[
5\sqrt{10}
\][/tex]
5. Combine the simplified terms:
Now, placing both terms together, we get:
[tex]\[
6\sqrt{5} - 5\sqrt{10}
\][/tex]
Thus, the expression [tex]\( \sqrt{5}(6 - 5\sqrt{2}) \)[/tex] simplifies to:
[tex]\[ 6\sqrt{5} - 5\sqrt{10} \][/tex]
Based on the provided options, this answer doesn't seem to match any of those exactly. However, you have already found the correct simplified form of the expression, which is the crucial part here. It’s important to check if the given options match any form of simplification provided.
### Final Answer:
The simplified form of the given expression is:
[tex]\[ 6\sqrt{5} - 5\sqrt{10} \][/tex]
None of the given options correspond directly to this expression, but this is the most simplified form of the given mathematical problem.
