Let's simplify the expression [tex]\(2 \sqrt{20} - 7 \sqrt{125} + 2 \sqrt{45}\)[/tex].
### Step 1: Simplify each square root term
First, we need to simplify each term under the square root.
#### Term 1: [tex]\(2 \sqrt{20}\)[/tex]
[tex]\[
2 \sqrt{20} = 2 \sqrt{4 \cdot 5} = 2 \cdot \sqrt{4} \cdot \sqrt{5} = 2 \cdot 2 \sqrt{5} = 4 \sqrt{5}
\][/tex]
#### Term 2: [tex]\(-7 \sqrt{125}\)[/tex]
[tex]\[
-7 \sqrt{125} = -7 \sqrt{25 \cdot 5} = -7 \cdot \sqrt{25} \cdot \sqrt{5} = -7 \cdot 5 \sqrt{5} = -35 \sqrt{5}
\][/tex]
#### Term 3: [tex]\(2 \sqrt{45}\)[/tex]
[tex]\[
2 \sqrt{45} = 2 \sqrt{9 \cdot 5} = 2 \cdot \sqrt{9} \cdot \sqrt{5} = 2 \cdot 3 \sqrt{5} = 6 \sqrt{5}
\][/tex]
### Step 2: Combine like terms
Now we combine the terms that contain [tex]\(\sqrt{5}\)[/tex]:
[tex]\[
4 \sqrt{5} - 35 \sqrt{5} + 6 \sqrt{5}
\][/tex]
Combine the coefficients of [tex]\(\sqrt{5}\)[/tex]:
[tex]\[
(4 - 35 + 6) \sqrt{5} = -25 \sqrt{5}
\][/tex]
Hence, the simplified expression is:
[tex]\[
-25 \sqrt{5}
\][/tex]
### Step 3: Numerical evaluation (to provide context and how precise the answer is)
To give a numerical evaluation:
- [tex]\(4 \sqrt{5} \approx 8.94427190999916\)[/tex]
- [tex]\(-35 \sqrt{5} \approx -78.26237921249265\)[/tex]
- [tex]\(6 \sqrt{5} \approx 13.416407864998739\)[/tex]
Thus, combining these numerically:
[tex]\[
8.94427190999916 - 78.26237921249265 + 13.416407864998739 \approx -55.90169943749474
\][/tex]
### Final Answer
The simplified form of [tex]\(2 \sqrt{20} - 7 \sqrt{125} + 2 \sqrt{45}\)[/tex] is:
[tex]\[
-25 \sqrt{5}
\][/tex]
And numerically, this evaluates to approximately:
[tex]\[
-55.90169943749474
\][/tex]
