Question

Which expression is equivalent to square root 25x^9y^3/64x^6y^11

Answer 1

Answer:

$\large\boxed{\dfrac{5x\sqrt{x}}{8y^4}}$

Step-by-step explanation:

$\sqrt{\dfrac{25x^9y^3}{64x^6y^{11}}}\qquad\text{use}\ \sqrt{ab}=\sqrt{a}\cdot\sqrt{b},\ \sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}};\ \dfrac{a^m}{a^n}=a^{m-n}\\\\=\dfrac{\sqrt{25}}{\sqrt{64}}\cdot \sqrt{x^{9-6}y^{3-11}}=\dfrac{5}{8}\sqrt{x^3y^{-8}}=\dfrac{5}{8}\sqrt{x^3}\cdot\sqrt{y^{-8}}\\\\\text{use}\ a^na^m=a^{n+m},\ (a^n)^m=a^{nm}\\\\=\dfrac{5}{8}\sqrt{x^{2+1}}\cdot\sqrt{y^{(-4)(2)}}=\dfrac{5}{8}\sqrt{x^2x}\cdot\sqrt{(y^{-4})^2}$

$=\dfrac{5}{8}\sqrt{x^2}\cdot\sqrt{x}\cdot\sqrt{(y^{-4})^2}\qquad\text{use}\ \sqrt{a^2}=a\ \text{for}\ a\geq0\\\\=\dfrac{5}{8}x\sqrt{x}\cdot y^{-4}\qquad\text{use}\ a^{-n}=\dfrac{1}{a^n}\\\\=\dfrac{5x\sqrt{x}}{8y^4}$