which expression is equivalent to $sqrt{\frac{25x^{9}y^{3}}{64x^{6}y^{11}}}$? assume $x > 0$ and $y >…

which expression is equivalent to $sqrt{\frac{25x^{9}y^{3}}{64x^{6}y^{11}}}$? assume $x > 0$ and $y > 0$.\n$\frac{8y^{4}sqrt{x}}{5x}$\n$\frac{8y^{2}sqrt{x}}{5}$\n$\frac{5sqrt{x}}{8y^{2}}$\n$\frac{5xsqrt{x}}{8y^{4}}$

which expression is equivalent to $sqrt{\frac{25x^{9}y^{3}}{64x^{6}y^{11}}}$? assume $x > 0$ and $y > 0$.\n$\frac{8y^{4}sqrt{x}}{5x}$\n$\frac{8y^{2}sqrt{x}}{5}$\n$\frac{5sqrt{x}}{8y^{2}}$\n$\frac{5xsqrt{x}}{8y^{4}}$

Answer

Explanation:

Step1: Simplify fraction under square - root

Use the quotient rule of exponents $\frac{a^m}{a^n}=a^{m - n}$ and $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$. $\sqrt{\frac{25x^{9}y^{3}}{64x^{6}y^{11}}}=\frac{\sqrt{25x^{9}y^{3}}}{\sqrt{64x^{6}y^{11}}}=\frac{5x^{\frac{9}{2}}y^{\frac{3}{2}}}{8x^{3}y^{\frac{11}{2}}}$

Step2: Simplify the fraction with exponents

Use the quotient rule of exponents $\frac{a^m}{a^n}=a^{m - n}$ again. $\frac{5x^{\frac{9}{2}}y^{\frac{3}{2}}}{8x^{3}y^{\frac{11}{2}}}=\frac{5}{8}x^{\frac{9}{2}-3}y^{\frac{3}{2}-\frac{11}{2}}=\frac{5}{8}x^{\frac{9 - 6}{2}}y^{\frac{3 - 11}{2}}=\frac{5}{8}x^{\frac{3}{2}}y^{-4}$

Step3: Rewrite with positive exponents

Use the rule $a^{-n}=\frac{1}{a^{n}}$. $\frac{5}{8}x^{\frac{3}{2}}y^{-4}=\frac{5x\sqrt{x}}{8y^{4}}$

Answer:

D. $\frac{5x\sqrt{x}}{8y^{4}}$