Sure! Let's simplify the expression [tex]\(5 \sqrt{92 x^3 y^2}\)[/tex] step-by-step.
1. Factorize the Radicand: First, we look inside the square root and factorize [tex]\(92 x^3 y^2\)[/tex].
[tex]\[
92 = 4 \times 23
\][/tex]
So, the expression becomes:
[tex]\[
5 \sqrt{4 \times 23 \times x^3 \times y^2}
\][/tex]
2. Separate the Factors Inside the Square Root: We can split the square root into separate square roots for each factor.
[tex]\[
5 \sqrt{4} \cdot \sqrt{23} \cdot \sqrt{x^3} \cdot \sqrt{y^2}
\][/tex]
3. Simplify the Square Roots of Perfect Squares: We notice that [tex]\(\sqrt{4}\)[/tex] and [tex]\(\sqrt{y^2}\)[/tex] are perfect squares.
[tex]\[
\sqrt{4} = 2 \quad \text{and} \quad \sqrt{y^2} = y
\][/tex]
Now the expression looks like:
[tex]\[
5 \cdot 2 \cdot \sqrt{23} \cdot \sqrt{x^3} \cdot y
\][/tex]
4. Combine Constants: Multiplying the constants together:
[tex]\[
5 \times 2 = 10
\][/tex]
So, we have:
[tex]\[
10 \sqrt{23} \cdot \sqrt{x^3} \cdot y
\][/tex]
5. Simplify the Remaining Square Roots: Notice that [tex]\(x^3\)[/tex] can be rewritten as [tex]\(x^2 \times x\)[/tex], allowing us to split it.
[tex]\[
\sqrt{x^3} = \sqrt{x^2 \times x} = \sqrt{x^2} \cdot \sqrt{x} = x \sqrt{x}
\][/tex]
Substituting back:
[tex]\[
10 \sqrt{23} \cdot x \sqrt{x} \cdot y
\][/tex]
6. Combine the Terms: Combine the terms back together:
[tex]\[
10 \sqrt{23} \cdot x \sqrt{x} \cdot y = 10 \sqrt{23} \cdot \sqrt{x^3} \cdot y
\][/tex]
7. Final Simplified Expression: The final, simplified form of the expression is:
[tex]\[
10 \sqrt{23} \sqrt{x^3 y^2}
\][/tex]
Therefore, the simplified form of [tex]\(5 \sqrt{92 x^3 y^2}\)[/tex] is:
[tex]\[
10 \sqrt{23} \cdot \sqrt{x^3 y^2}
\][/tex]
