Chapter 10, Problem 1CR

To determine

To determine, for positive $x$ and $y$ , which expression is equivalent to ${\left(x^{\frac{1}{2}}-y^{\frac{1}{2}}\right)}^2$

Answer to Problem 1CR

Option b) $x+2\sqrt{xy}+y$ is equivalent to ${\left(x^{\frac{1}{2}}-y^{\frac{1}{2}}\right)}^2$

Explanation of Solution

Given information:

Given expression is ${\left(x^{\frac{1}{2}}-y^{\frac{1}{2}}\right)}^2$

Here we shall use the identity ${\left(a+b\right)}^2=a^2+b^2+2ab$

Now we know that $a^{\frac{1}{2}}=\sqrt{a}$

Hence we solve and apply the identity we get,

  $\begin{array}{c}{\left(\sqrt{x}+\sqrt{y}\right)}^2={\left(\sqrt{x}\right)}^2+{\left(\sqrt{y}\right)}^2+2\sqrt{xy}\\ =x+2\sqrt{xy}+y\end{array}$