1. Stating the problem: We need to simplify each expression under the square root, breaking it down into factors with perfect squares and variables with even powers so we can simplify the root. 2. Simplify each expression: a. $\sqrt{50 n^2}$ - Factor 50 as $25 \times 2$. - Use $\sqrt{a b} = \sqrt{a} \sqrt{b}$: $\sqrt{25 \times 2 \times n^2} = \sqrt{25} \times \sqrt{2} \times \sqrt{n^2}$ - Simplify perfect squares: $5 \times \sqrt{2} \times n = 5 n \sqrt{2}$ b. $\sqrt{72 n^7}$ - Factor 72 as $36 \times 2$. - Split variables: $n^7 = n^6 \times n$ (since 6 is even power) - $\sqrt{72 n^7} = \sqrt{36} \times \sqrt{2} \times \sqrt{n^6} \times \sqrt{n} = 6 \times \sqrt{2} \times n^3 \times \sqrt{n} = 6 n^3 \sqrt{2 n}$ c. $\sqrt{18 x^3}$ - Factor 18 as $9 \times 2$. - Write $x^3 = x^2 \times x$. - $\sqrt{18 x^3} = \sqrt{9} \times \sqrt{2} \times \sqrt{x^2} \times \sqrt{x} = 3 \times \sqrt{2} \times x \times \sqrt{x} = 3 x \sqrt{2 x}$ d. $\sqrt{98 x^5 y^6}$ - Factor 98 as $49 \times 2$. - Split powers: $x^5 = x^4 \times x$, $y^6$ is already an even power. - $\sqrt{98 x^5 y^6} = \sqrt{49} \times \sqrt{2} \times \sqrt{x^4} \times \sqrt{x} \times \sqrt{y^6} = 7 \times \sqrt{2} \times x^2 \times \sqrt{x} \times y^3 = 7 x^2 y^3 \sqrt{2 x}$ e. $\sqrt{320 a b^4}$ - Factor 320 as $64 \times 5$. - $b^4$ is even power. - $\sqrt{320 a b^4} = \sqrt{64} \times \sqrt{5} \times \sqrt{a} \times \sqrt{b^4} = 8 \times \sqrt{5} \times \sqrt{a} \times b^2 = 8 b^2 \sqrt{5 a}$ Final answers: a. $5 n \sqrt{2}$ b. $6 n^3 \sqrt{2 n}$ c. $3 x \sqrt{2 x}$ d. $7 x^2 y^3 \sqrt{2 x}$ e. $8 b^2 \sqrt{5 a}$