Question

simplify square root of 80x^3y^2

Answer 1

To simplify the expression \(\sqrt{80x^3y^2}\), we can break it down into its components.

1. Factor \(80\):
\[
80 = 16 \times 5 = 4^2 \times 5
\]

2. Write \(x^3\) and \(y^2\) in a form suitable for taking square roots:
- \(x^3 = x^2 \cdot x\)
- \(y^2\) is already a perfect square.

3. Now, rewrite the square root:
\[
\sqrt{80x^3y^2} = \sqrt{16 \cdot 5 \cdot x^2 \cdot x \cdot y^2}
\]

4. Apply the property of square roots:
\[
\sqrt{80x^3y^2} = \sqrt{16} \cdot \sqrt{5} \cdot \sqrt{x^2} \cdot \sqrt{x} \cdot \sqrt{y^2}
\]

5. Calculate the square roots of perfect squares:
- \(\sqrt{16} = 4\)
- \(\sqrt{x^2} = x\)
- \(\sqrt{y^2} = y\)

6. Combine the results:
\[
\sqrt{80x^3y^2} = 4 \cdot \sqrt{5} \cdot x \cdot \sqrt{x} \cdot y
\]
\[
= 4xy\sqrt{5x}
\]

Thus, the simplified form of \(\sqrt{80x^3y^2}\) is:
\[
\boxed{4xy\sqrt{5x}}
\]