- #1

RChristenk

- 51

- 5

- Homework Statement
- Rationalize the denominator: ##\sqrt{\dfrac{1}{2x^3y^5}}##

- Relevant Equations
- Operations involving radicals

##\sqrt{\dfrac{1}{2x^3y^5}}=\dfrac{1}{\sqrt{2\cdot x^2 \cdot x \cdot y^2 \cdot y^2 \cdot y}}=\dfrac{1}{|x|\cdot |y|\cdot |y| \cdot \sqrt{2xy}}=\dfrac{1}{|x|y^2\sqrt{2xy}}##

##\Rightarrow \dfrac{1}{|x|y^2\sqrt{2xy}} \cdot \dfrac{\sqrt{2xy}}{\sqrt{2xy}}=\dfrac{\sqrt{2xy}}{|x|y^2 \cdot 2xy}=\dfrac{\sqrt{2xy}}{2|x|xy^3}##

But the solution is given as ##\sqrt{\dfrac{1}{2x^3y^5}}=\dfrac{1}{xy^2\sqrt{2xy}}\cdot \dfrac{\sqrt{2xy}}{\sqrt{2xy}}=\dfrac{\sqrt{2xy}}{2x^2y^3}## without any consideration for the absolute value. But the definition is ##\sqrt{x^2}=|x|##, so I'm not understanding why the book solution ignores this.

##\Rightarrow \dfrac{1}{|x|y^2\sqrt{2xy}} \cdot \dfrac{\sqrt{2xy}}{\sqrt{2xy}}=\dfrac{\sqrt{2xy}}{|x|y^2 \cdot 2xy}=\dfrac{\sqrt{2xy}}{2|x|xy^3}##

But the solution is given as ##\sqrt{\dfrac{1}{2x^3y^5}}=\dfrac{1}{xy^2\sqrt{2xy}}\cdot \dfrac{\sqrt{2xy}}{\sqrt{2xy}}=\dfrac{\sqrt{2xy}}{2x^2y^3}## without any consideration for the absolute value. But the definition is ##\sqrt{x^2}=|x|##, so I'm not understanding why the book solution ignores this.