Sure, let's simplify the expression [tex]\(\sqrt{12 x^3 y^2}\)[/tex] step by step.
1. Break down the expression under the square root:
We start with the expression inside the square root:
[tex]\[
12 x^3 y^2
\][/tex]
2. Factor out the constant and the variables separately:
- [tex]\(12\)[/tex] can be factored as [tex]\(4 \times 3\)[/tex].
- The variable expression [tex]\(x^3\)[/tex] remains as it is.
- The variable expression [tex]\(y^2\)[/tex] remains as it is.
So, we can rewrite [tex]\(12 x^3 y^2\)[/tex] as:
[tex]\[
12 x^3 y^2 = 4 \cdot 3 \cdot x^3 \cdot y^2
\][/tex]
3. Apply the square root to each factor separately:
[tex]\[
\sqrt{12 x^3 y^2} = \sqrt{4 \cdot 3 \cdot x^3 \cdot y^2}
\][/tex]
Using the property [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex], we get:
[tex]\[
\sqrt{4 \cdot 3 \cdot x^3 \cdot y^2} = \sqrt{4} \cdot \sqrt{3} \cdot \sqrt{x^3} \cdot \sqrt{y^2}
\][/tex]
4. Simplify the square roots of each factor:
- [tex]\(\sqrt{4} = 2\)[/tex]
- [tex]\(\sqrt{3}\)[/tex] remains as it is because [tex]\(3\)[/tex] is not a square number.
- [tex]\(\sqrt{x^3}\)[/tex] can be written as [tex]\(x^{\frac{3}{2}}\)[/tex] which we can further simplify as [tex]\(x^{1} \cdot x^{\frac{1}{2}}\)[/tex] or [tex]\(x\sqrt{x}\)[/tex].
- [tex]\(\sqrt{y^2} = y\)[/tex]
So, we have:
[tex]\[
\sqrt{4} \cdot \sqrt{3} \cdot \sqrt{x^3} \cdot \sqrt{y^2} = 2 \cdot \sqrt{3} \cdot x \sqrt{x} \cdot y
\][/tex]
5. Combine the simplified parts:
Now we combine all the simplified parts together:
[tex]\[
2 \cdot \sqrt{3} \cdot x \sqrt{x} \cdot y
\][/tex]
Which gives us the final simplified expression:
[tex]\[
\sqrt{12 x^3 y^2} = 2\sqrt{3} \cdot x \sqrt{x} \cdot y
\][/tex]
Thus, the simplest form of [tex]\(\sqrt{12 x^3 y^2}\)[/tex] is:
[tex]\[
2 \cdot \sqrt{3} \cdot \sqrt{x^3 \cdot y^2}
\][/tex]
And this is the most simplified form of the given expression.
