Question

simplify the following expression.
\sqrt{x^{7}y^{7}} \cdot 2\sqrt{5x^{3}y^{2}}
\sqrt{x^{7}y^{7}} \cdot 2\sqrt{5x^{3}y^{2}} = \square
(simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

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

First, we can combine the two square roots using the property \( \sqrt{a} \cdot \sqrt{b}=\sqrt{ab} \). Also, we have a coefficient of 2, so we can rewrite the expression as:
\( 2\sqrt{x^{7}y^{7}\cdot5x^{3}y^{2}} \)

Step2: Combine like terms (exponents)

When multiplying variables with exponents, we add the exponents. For \( x \): \( 7 + 3=10 \), for \( y \): \( 7+ 2 = 9 \), and the constant is 5. So the expression inside the square root becomes:
\( 2\sqrt{5x^{10}y^{9}} \)

Step3: Simplify the square root

We know that \( \sqrt{x^{10}}=x^{5} \) (since \( (x^{5})^{2}=x^{10} \)) and \( \sqrt{y^{9}}=\sqrt{y^{8}\cdot y}=y^{4}\sqrt{y} \) (since \( (y^{4})^{2}=y^{8} \)). So we can split the square root:
\( 2\cdot x^{5}\cdot y^{4}\cdot\sqrt{5y} \)

Step4: Multiply the coefficients (just 2 here)

There are no other coefficients to multiply, so we just combine the terms:
\( 2x^{5}y^{4}\sqrt{5y} \)

Answer:

\( 2x^{5}y^{4}\sqrt{5y} \)