Question

simplify the following expression.
\sqrt{x^{7}y^{7}} \cdot 2\sqrt{5x^{3}y^{10}}

Explanation:

Step1: Use the property of square roots \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \)

First, combine the two square root terms. The coefficient 2 remains outside. So we have \( 2\sqrt{x^{7}y^{7} \cdot 5x^{3}y^{10}} \)

Step2: Combine like terms in the radicand

For the \( x \) terms: \( x^{7} \cdot x^{3}=x^{7 + 3}=x^{10} \)
For the \( y \) terms: \( y^{7} \cdot y^{10}=y^{7+10}=y^{17} \)
And the constant factor is 5. So the radicand becomes \( 5x^{10}y^{17} \), and the expression is \( 2\sqrt{5x^{10}y^{17}} \)

Step3: Simplify the square root

We know that \( \sqrt{x^{10}}=x^{5} \) (since \( (x^{5})^{2}=x^{10} \)) and \( \sqrt{y^{17}}=\sqrt{y^{16}\cdot y}=y^{8}\sqrt{y} \) (since \( (y^{8})^{2}=y^{16} \))
So we can rewrite the square root as \( \sqrt{5x^{10}y^{17}}=\sqrt{5}\cdot\sqrt{x^{10}}\cdot\sqrt{y^{17}} = x^{5}y^{8}\sqrt{5y} \)

Step4: Multiply by the coefficient 2

Multiply the simplified square root by 2: \( 2\times x^{5}y^{8}\sqrt{5y}=2x^{5}y^{8}\sqrt{5y} \)

Answer:

\( 2x^{5}y^{8}\sqrt{5y} \)