### Thinking about Algebra

Package of problems

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## Problem

$\sqrt{x}$ or $x^{\frac{1}{2}}$? Whichever we prefer, we need to be comfortable working with indices and roots in either form.

Below is a series of equations, for which you should think about the following questions.

For what values of $x$ and $y$ do these statements make sense?

What are the possible values for the expressions?

1. $(xy)^\frac{1}{2}=x^\frac{1}{2}y^\frac{1}{2}$

2. $(xy)^{\frac{5}{3}}=x^{\frac{5}{3}}y^{\frac{5}{3}}$

3. $(xy)^{\frac{2}{3}}=x^{\frac{2}{3}}y^{\frac{2}{3}}$

4. $(xy)^{-\frac{1}{2}}= x^{-\frac{1}{2}}y^{-\frac{1}{2}}$

5. $(xy)^\frac{1}{2} = x^\frac{1}{3}y^\frac{2}{3}$

6. $(xy)^{-2}= x^{2}y^{2}$

7. $\left(\frac{x}{y}\right)^{-\frac{1}{3}} = x^{-\frac{1}{3}}y^{\frac{1}{3}}$

8. $\left(\frac{x}{y}-\frac{1}{y}\right)^{\frac{1}{2}}= y^{-\frac{1}{2}}(x-1)^\frac{1}{2}$