Question

Simplify square root with variables
√50x^4 y^5

Answer 1

To simplify the expression \(\sqrt{50x^4 y^5}\), we can break it down into separate square roots for the numerical and variable parts.

1. First, factor the number and express it in terms of perfect squares:
\[
50 = 25 \cdot 2 = 5^2 \cdot 2
\]

2. Next, we can rewrite the variable parts:
- For \(x^4\), since \(x^4 = (x^2)^2\), we can take the square root.
- For \(y^5\), we can rewrite it as \(y^4 \cdot y = (y^2)^2 \cdot y\), allowing us to take the square root of the \(y^4\) part.

Now we can rewrite the expression:
\[
\sqrt{50x^4 y^5} = \sqrt{50} \cdot \sqrt{x^4} \cdot \sqrt{y^5}
\]

This simplifies to:
\[
= \sqrt{25 \cdot 2} \cdot x^2 \cdot \sqrt{y^4 \cdot y}
\]
\[
= \sqrt{25} \cdot \sqrt{2} \cdot x^2 \cdot \sqrt{y^4} \cdot \sqrt{y}
\]
\[
= 5 \cdot \sqrt{2} \cdot x^2 \cdot y^2 \cdot \sqrt{y}
\]

Putting it all together, we have:
\[
= 5x^2y^2\sqrt{2y}
\]

Thus, the simplified form of \(\sqrt{50x^4 y^5}\) is:
\[
\boxed{5x^2y^2\sqrt{2y}}
\]