Remember that $k^{1/2} = \sqrt{k}$ and $k^{3/2} = k\sqrt{k}$. Try rewriting the numerator using square roots and see if a common factor appears.

After factoring the numerator, check whether the factor $k - 1$ appears. If it does, think about what happens when a common factor in the numerator and denominator is canceled, given that $k \ne 1$.

In Desmos, type the expression using $x$ instead of $k$:

(x^(3/2) - x^(1/2))/(x - 1)

This is the graph or value of the original expression for $x > 0$ and $x \ne 1$.

On new lines, type each option (again using $x$):

1. x
2. (x - 1)*sqrt(x)
3. 1/sqrt(x)
4. sqrt(x)

These represent the four possible simplified forms.

Either:

• Look at the graphs: see which of the four option graphs lies exactly on top of the graph of (x^(3/2) - x^(1/2))/(x - 1) for $x > 0$, $x \ne 1$; or
• Use a table (click the gear and add a table) to compare numeric values at a few positive $x$ values not equal to 1 (such as $x=4$ or $x=9$).

The option whose values always match the original expression for allowed $x$ is the correct equivalent expression.

First, recognize that $k^{1/2} = \sqrt{k}$ and $k^{3/2} = k\sqrt{k}$.

So the numerator becomes

Now the original expression can be written with this factored numerator.

Substitute the factored numerator into the fraction:

Because $k \ne 1$, the factor $k - 1$ is not zero, so it can be canceled as a common factor of the numerator and denominator:

Since $\dfrac{k - 1}{k - 1} = 1$ for $k \ne 1$, the entire expression simplifies to

Among the choices, this corresponds to option D.) $\sqrt{k}$.