Simplify: \( \sqrt{40 x^{6} y^{7}} \)

To simplify \( \sqrt{40 x^{6} y^{7}} \), we can break it down into its factors. First, we rewrite \( 40 \) as \( 4 \times 10 \), and we know that \( 4 \) is a perfect square. Now, we can separate the square root: \[ \sqrt{40 x^{6} y^{7}} = \sqrt{4 \times 10} \times \sqrt{x^{6}} \times \sqrt{y^{7}} \] Calculating each part: 1. \( \sqrt{4} = 2 \) 2. \( \sqrt{x^{6}} = x^{3} \) (since \( x^{6} = (x^{3})^2 \)) 3. For \( y^{7} \), we can rewrite it as \( y^{6} \times y \): \( \sqrt{y^{7}} = \sqrt{y^{6} \times y} = \sqrt{y^{6}} \times \sqrt{y} = y^{3} \sqrt{y} \) Putting it all together gives: \[ \sqrt{40 x^{6} y^{7}} = 2 \sqrt{10} \times x^{3} \times (y^{3} \sqrt{y}) \] Thus, the final simplified expression is: \[ 2 x^{3} y^{3} \sqrt{10y} \]