Let's break down and simplify the expression [tex]\(2 \sqrt{75 x y^3}\)[/tex] step by step.
1. Factorize the number inside the square root:
The number 75 can be factorized as [tex]\(75 = 25 \times 3\)[/tex].
2. Rewrite the square root expression:
[tex]\[
2 \sqrt{75 x y^3} = 2 \sqrt{(25 \times 3) x y^3}
\][/tex]
3. Separate the product inside the square root:
[tex]\[
2 \sqrt{25 \times 3 x y^3} = 2 \sqrt{25} \cdot \sqrt{3 x y^3}
\][/tex]
4. Simplify the square roots:
- [tex]\(\sqrt{25}\)[/tex] is 5.
Therefore,
[tex]\[
2 \sqrt{25} \cdot \sqrt{3 x y^3} = 2 \times 5 \cdot \sqrt{3 x y^3} = 10 \sqrt{3 x y^3}
\][/tex]
5. Simplify the square root involving the variables:
Note that [tex]\(y^3\)[/tex] can be written as [tex]\(y^2 \times y\)[/tex]. Thus,
[tex]\[
\sqrt{3 x y^3} = \sqrt{3 x y^2 \cdot y} = \sqrt{3 x y^2} \cdot \sqrt{y}
\][/tex]
- [tex]\(\sqrt{y^2}\)[/tex] is [tex]\(y\)[/tex].
Therefore,
[tex]\[
\sqrt{3 x y^2 \cdot y} = \sqrt{3 x} \cdot y \cdot \sqrt{y} = y \cdot \sqrt{3 x y}
\][/tex]
6. Combine the results:
Plugging this back into our simplified expression, we get:
[tex]\[
10 \sqrt{3 x y^3} = 10 y \sqrt{3 x y}
\][/tex]
Thus, the simplified form of [tex]\(2 \sqrt{75 x y^3}\)[/tex] is:
[tex]\[
10 y \sqrt{3 x y}
\][/tex]
