\(\sqrt{x}\) 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?

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

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

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

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

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

\((xy)^{-2}= x^{2}y^{2}\)

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

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