Question

simplify the following expression.\\(\sqrt{x^{5}y^{5}}\cdot 2\sqrt{5x^{9}y^{8}}\\)\\(\sqrt{x^{5}y^{5}}\cdot 2\sqrt{5x^{9}y^{8}}=\square\\)\\((\text{simplify your answer. type an exact answer, using}\\)

Explanation:

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

First, combine the two square roots. The coefficient 2 remains outside. So we have \( 2\sqrt{x^{5}y^{5}\cdot5x^{9}y^{8}} \)

Step2: Combine like terms in the radicand

For the \( x \) terms: \( x^{5} \cdot x^{9}=x^{5 + 9}=x^{14} \)
For the \( y \) terms: \( y^{5}\cdot y^{8}=y^{5 + 8}=y^{13} \)
For the constant term: we have 5. So the radicand becomes \( 5x^{14}y^{13} \)
Now the expression is \( 2\sqrt{5x^{14}y^{13}} \)

Step3: Simplify the square root

We know that \( \sqrt{x^{14}}=x^{7} \) (since \( (x^{7})^{2}=x^{14} \))
For \( y^{13} \), we can write \( y^{13}=y^{12}\cdot y=(y^{6})^{2}\cdot y \), so \( \sqrt{y^{13}}=y^{6}\sqrt{y} \)
So \( \sqrt{5x^{14}y^{13}}=x^{7}y^{6}\sqrt{5y} \)

Step4: Multiply by the coefficient 2

Multiply 2 with \( x^{7}y^{6}\sqrt{5y} \), we get \( 2x^{7}y^{6}\sqrt{5y} \)

Answer:

\( 2x^{7}y^{6}\sqrt{5y} \)