Question 171·Hard·Equivalent Expressions

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For $k>0$ and $k\ne 1$ , which expression is equivalent to

\frac{k^{3/2}-k^{1/2}}{k-1}?

For rational expressions with fractional exponents, first convert fractional powers to roots when helpful (for example, $k^{1/2} = \sqrt{k}$ , $k^{3/2} = k\sqrt{k}$ ), then look for common factors in the numerator and denominator. Factor out the greatest common factor, cancel any common nonzero factors (using given domain restrictions like $k > 0$ , $k \ne 1$ to justify cancellation), and only then match the fully simplified form to the answer choices. Avoid misusing exponent rules on sums or differences—those rules apply only to products and quotients with the same base.

Hints

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Look for a common factor in the numerator

Both terms in the numerator, $k^{3/2}$ and $k^{1/2}$ , involve the same base $k$ . Can you factor out a common power of $k$ from these two terms?

Rewrite fractional exponents as roots

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.

Use factoring to cancel with the denominator

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 $k \ne 1$ .

Desmos Guide

Enter the original expression

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 $x \ne 1$ .

Enter each answer choice as a separate expression

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

x
(x - 1)*sqrt(x)
1/sqrt(x)
sqrt(x)

These represent the four possible simplified forms.

Compare the original with the choices

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.

Step-by-step Explanation

Rewrite and factor the numerator

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

So the numerator becomes

k^{3/2} - k^{1/2} = k\sqrt{k} - \sqrt{k} = \sqrt{k}(k - 1).

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

Write the fraction using the factored form

Substitute the factored numerator into the fraction:

\frac{k^{3/2} - k^{1/2}}{k - 1} = \frac{\sqrt{k}(k - 1)}{k - 1}.

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:

\frac{\sqrt{k}(k - 1)}{k - 1} = \sqrt{k} \cdot \frac{k - 1}{k - 1}.

Simplify and match the answer choice

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

\sqrt{k}.

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

Question 171·Hard·Equivalent Expressions

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation