Question

assuming $x$ and $y$ are both positive, write the following expression in simplest radical form.
$y^{2}\sqrt{25x^{7}y^{7}}$

Explanation:

Step1: Simplify the square root of the perfect squares

First, we can break down the terms inside the square root. We know that \( \sqrt{25} = 5 \), \( \sqrt{x^7} = \sqrt{x^6 \cdot x}=x^3\sqrt{x} \) (since \( x^6=(x^3)^2 \) and using the property \( \sqrt{ab}=\sqrt{a}\cdot\sqrt{b} \) for \( a,b\geq0 \)), and \( \sqrt{y^7}=\sqrt{y^6\cdot y}=y^3\sqrt{y} \) (similarly, \( y^6=(y^3)^2 \)). So, \( \sqrt{25x^7y^7}=\sqrt{25}\cdot\sqrt{x^7}\cdot\sqrt{y^7}=5\cdot x^3\sqrt{x}\cdot y^3\sqrt{y} = 5x^3y^3\sqrt{xy} \).

Step2: Multiply by the outside term

Now we have the original expression \( y^2\sqrt{25x^7y^7} \), and we substitute the simplified square root from Step 1. So we multiply \( y^2 \) with \( 5x^3y^3\sqrt{xy} \). Using the property of exponents \( a^m\cdot a^n = a^{m + n} \) for the \( y \)-terms: \( y^2\cdot y^3=y^{2 + 3}=y^5 \). So the expression becomes \( 5x^3y^5\sqrt{xy} \).

Answer:

\( 5x^{3}y^{5}\sqrt{xy} \)