To simplify the given expression [tex]\( 5 v \sqrt{45 y^5} - y^2 \sqrt{80 y v^2} \)[/tex], let's handle each term separately and then combine them.
### Step 1: Simplify [tex]\( 5 v \sqrt{45 y^5} \)[/tex]
1. Factor Inside the Square Root:
[tex]\[ 45 y^5 = 9 \cdot 5 \cdot y^4 \cdot y = 9y^4 \cdot 5y \][/tex]
2. Break Down the Square Root:
[tex]\[ \sqrt{45 y^5} = \sqrt{9 y^4 \cdot 5 y} \][/tex]
[tex]\[ \sqrt{9 y^4 \cdot 5 y} = \sqrt{9 y^4} \cdot \sqrt{5 y} \][/tex]
3. Simplify Each Part:
[tex]\[ \sqrt{9 y^4} = 3 y^2 \][/tex]
[tex]\[ \sqrt{5 y} \text{ remains as it is} \][/tex]
4. Combine the Simplified Parts with the Coefficient:
[tex]\[ 5 v \sqrt{45 y^5} = 5 v \cdot 3 y^2 \cdot \sqrt{5 y} = 15 v y^2 \sqrt{5 y} \][/tex]
### Step 2: Simplify [tex]\( y^2 \sqrt{80 y v^2} \)[/tex]
1. Factor Inside the Square Root:
[tex]\[ 80 y v^2 = 16 \cdot 5 \cdot y \cdot v^2 = 16v^2 \cdot 5y \][/tex]
2. Break Down the Square Root:
[tex]\[ \sqrt{80 y v^2} = \sqrt{16 v^2 \cdot 5 y} \][/tex]
[tex]\[ \sqrt{16 v^2 \cdot 5 y} = \sqrt{16 v^2} \cdot \sqrt{5 y} \][/tex]
3. Simplify Each Part:
[tex]\[ \sqrt{16 v^2} = 4 v \][/tex]
[tex]\[ \sqrt{5 y} \text{ remains as it is} \][/tex]
4. Combine the Simplified Parts with the Coefficient:
[tex]\[ y^2 \sqrt{80 y v^2} = y^2 \cdot 4 v \cdot \sqrt{5 y} = 4 v y^2 \sqrt{5 y} \][/tex]
### Step 3: Combine the Simplified Terms
Now, we combine the simplified terms:
[tex]\[ 15 v y^2 \sqrt{5 y} - 4 v y^2 \sqrt{5 y} \][/tex]
Both terms contain the common factor [tex]\( v y^2 \sqrt{5 y} \)[/tex]:
[tex]\[ (15 v y^2 \sqrt{5 y}) - (4 v y^2 \sqrt{5 y}) = (15 - 4) v y^2 \sqrt{5 y} \][/tex]
[tex]\[ = 11 v y^2 \sqrt{5 y} \][/tex]
### Final Simplified Expression
The simplified expression is:
[tex]\[ \boxed{11 v y^2 \sqrt{5 y}} \][/tex]
