Let's solve the expression [tex]\(3 y^2 \sqrt{18 x^2 y^5}\)[/tex] step-by-step by simplifying it:
1. Simplify inside the square root:
[tex]\[
18 x^2 y^5
\][/tex]
Factorize each component inside the square root:
[tex]\[
18 = 2 \cdot 3^2
\][/tex]
So we can express this as:
[tex]\[
18 x^2 y^5 = (2 \cdot 3^2) x^2 y^5
\][/tex]
2. Use properties of square roots:
Recall the property:
[tex]\[
\sqrt{a \cdot b \cdot c} = \sqrt{a} \cdot \sqrt{b} \cdot \sqrt{c}
\][/tex]
Apply it to break down the expression inside the square root:
[tex]\[
\sqrt{18 x^2 y^5} = \sqrt{2 \cdot 3^2 \cdot x^2 \cdot y^5}
= \sqrt{2} \cdot \sqrt{3^2} \cdot \sqrt{x^2} \cdot \sqrt{y^5}
\][/tex]
3. Evaluate individual square roots:
[tex]\[
\sqrt{3^2} = 3,\quad \sqrt{x^2} = x,\quad \sqrt{y^5} = \sqrt{y^4 \cdot y} = \sqrt{y^4} \cdot \sqrt{y} = y^2 \cdot \sqrt{y}
\][/tex]
Therefore:
[tex]\[
\sqrt{18 x^2 y^5} = \sqrt{2} \cdot 3 \cdot x \cdot y^2 \cdot \sqrt{y}
\][/tex]
4. Combine the simplified components:
Substitute back into the original expression:
[tex]\[
3 y^2 \sqrt{18 x^2 y^5} = 3 y^2 (3 \sqrt{2} x y^2 \sqrt{y})
\][/tex]
5. Simplify the overall expression:
Initially, distribute the multiplication:
[tex]\[
3 y^2 \cdot 3 \sqrt{2} x y^2 \sqrt{y} = 9 y^2 \cdot \sqrt{2} x y^2 \sqrt{y}
\][/tex]
Combine the terms involving [tex]\(y\)[/tex]:
[tex]\[
9 \sqrt{2} x y^2 \cdot y^2 \cdot \sqrt{y} = 9 \sqrt{2} x y^{2+2+\frac{1}{2}} = 9 \sqrt{2} x y^{4.5}
\][/tex]
So, the simplified expression is:
[tex]\[
9 \sqrt{2} x y^{4.5}
\][/tex]
Or, equivalently:
[tex]\[
9 \sqrt{2} x y^{\frac{9}{2}}
\][/tex]
