To simplify \(\sqrt{8x^3y^2}\), we start by factoring out the perfect squares from inside the square root.
- First, we can factor \(8\) as \(4 \times 2\), which gives us the perfect square \(4\).
- Next, we can separate the powers of \(x\) and \(y\):
- \(x^3\) can be written as \(x^2 \cdot x\), where \(x^2\) is a perfect square.
- \(y^2\) is already a perfect square.
Now we can rewrite the expression under the square root:
\[ \sqrt{8x^3y^2} = \sqrt{4 \cdot 2 \cdot x^2 \cdot x \cdot y^2} \]
Next, we use the property of square roots that states \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\):
\[ \sqrt{8x^3y^2} = \sqrt{4} \cdot \sqrt{2} \cdot \sqrt{x^2} \cdot \sqrt{x} \cdot \sqrt{y^2} \]
We know the square roots of the perfect squares:
- \(\sqrt{4} = 2\)
- \(\sqrt{x^2} = x\)
- \(\sqrt{y^2} = y\)
Putting it all together, we have:
\[ \sqrt{8x^3y^2} = 2 \cdot \sqrt{2} \cdot x \cdot \sqrt{x} \cdot y \]
Rearranging it, we get:
\[ \sqrt{8x^3y^2} = 2xy\sqrt{2x} \]
Thus, the simplified form is:
\[ \boxed{2xy\sqrt{2x}} \]
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