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Question 24·Hard·Equivalent Expressions

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Which expression is equivalent to $\dfrac{8r}{s}-\dfrac{18r^3}{s^3}$ , where $r\ne 0$ and $s\ne 0$ ?

For algebraic "equivalent expression" questions, first create a single fraction by finding a common denominator and combining the numerators. Then factor the resulting numerator completely, watching for common factors and patterns like differences of squares or perfect square trinomials. Finally, match your fully factored form—both numerator structure and denominator power—exactly to one of the choices, and use quick mental checks (like plugging in simple values) if you need extra confirmation.

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Make the denominators match

Before you try to factor anything, rewrite both fractions so they have the same denominator. What denominator do you want both terms to have?

Combine into a single fraction

Once the denominators match, combine the two fractions into one by subtracting the numerators over the common denominator.

Factor the numerator

Look at the combined numerator. What common factor can you pull out? After that, can you recognize a pattern like a difference of squares?

Compare with the answer choices

Pay attention to both the numerator and the power of $s$ in the denominator. Only one choice will match your fully factored form exactly.

Desmos Guide

Enter the original expression

Type (8*r)/s - (18*r^3)/(s^3) into Desmos. Then, to test, pick specific nonzero values for $r$ and $s$ (for example, r = 1, s = 2) and substitute them into the expression to see its numerical value.

Test each answer choice numerically

For the same values of $r$ and $s$ , type each choice into Desmos (for example, (2*r*(2*s-3*r))/(s^2) for choice A, etc.) and compare their numerical outputs to the original expression. Repeat with a different pair of nonzero values for $r$ and $s$ to be sure.

Identify the matching expression

The correct choice will give exactly the same numerical result as the original expression for all tested nonzero values of $r$ and $s$ . The other choices will disagree for at least one test pair.

Step-by-step Explanation

Get a common denominator

The expression is

\frac{8r}{s} - \frac{18r^3}{s^3}.

To combine these into one fraction, rewrite $\dfrac{8r}{s}$ so it has denominator $s^3$ .

Since $s \cdot s^2 = s^3$ , multiply the top and bottom of $\dfrac{8r}{s}$ by $s^2$ :

\frac{8r}{s} = \frac{8r s^2}{s^3}.

Now the expression is

\frac{8r s^2}{s^3} - \frac{18r^3}{s^3}.

Combine the fractions

Now that both fractions have denominator $s^3$ , subtract the numerators:

\frac{8r s^2}{s^3} - \frac{18r^3}{s^3} = \frac{8r s^2 - 18r^3}{s^3}.

So the whole expression is

\frac{8r s^2 - 18r^3}{s^3}.

Factor the numerator step by step

Look at the numerator $8r s^2 - 18r^3$ .

First, factor out the greatest common factor.

Both terms share $2r$ :

8r s^2 - 18r^3 = 2r(4s^2 - 9r^2).

Next, notice $4s^2 - 9r^2$ is a difference of squares:

$4s^2 = (2s)^2$
$9r^2 = (3r)^2$

4s^2 - 9r^2 = (2s - 3r)(2s + 3r).

Putting this into the factored numerator gives

8r s^2 - 18r^3 = 2r(2s - 3r)(2s + 3r).

Write the fully factored expression and match the choice

Substitute the factored numerator back over the denominator $s^3$ :

\frac{8r s^2 - 18r^3}{s^3} = \frac{2r(2s - 3r)(2s + 3r)}{s^3}.

This matches answer choice D, so the equivalent expression is $\dfrac{2r(2s-3r)(2s+3r)}{s^3}$ .

Question 24·Hard·Equivalent Expressions

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Step-by-step Explanation

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Step-by-step Explanation