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SAT Equivalent Expressions Question 179 | Free SAT Prep | Aniko

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Question 179·Medium·Equivalent Expressions

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Which expression is equivalent to

\frac{x^{3/2}-9x^{1/2}}{\sqrt{x}-3},

where $x \ge 0$ and $x \ne 9$ ?

For rational expressions with fractional exponents, first rewrite the exponents using roots (for example, $x^{1/2}$ as $\sqrt{x}$ ) to make patterns easier to see. Then factor out any common terms in the numerator and look for special factoring patterns like the difference of squares so you can cancel a common binomial factor with the denominator. Always check that you are not canceling at a point where the denominator is zero, and only then distribute any remaining factors and match the simplified form to the answer choices.

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Connect fractional exponents and square roots

Rewrite $x^{1/2}$ and $x^{3/2}$ using square root notation. What does $x^{1/2}$ equal in terms of $\sqrt{x}$ ?

Factor the numerator

Look at $x^{3/2}-9x^{1/2}$ . Both terms contain $x^{1/2}$ . Try factoring $x^{1/2}$ out of the numerator.

Recognize a difference of squares

After factoring out $x^{1/2}$ , you will see $x-9$ . Think of $x$ as $(\sqrt{x})^2$ . How can you factor $x-9$ using $\sqrt{x}$ and 3?

Desmos Guide

Enter the original expression

Type the original expression as ((x^(3/2)) - 9x^(1/2))/(sqrt(x) - 3) and restrict the domain to x >= 0 and x != 9 (for example, you can add a restriction like {x>=0, x!=9} in Desmos).

Enter each answer choice as a separate function

On new lines, enter y = x + 3sqrt(x), y = x - 3sqrt(x), y = sqrt(x) + 3, and y = x - 9. These represent the four choices.

Compare graphs or values

Either compare the graphs visually (which one lies exactly on top of the original expression’s graph for $x \ge 0$ , $x \ne 9$ ?) or pick a value like $x=4$ or $x=16$ (not 9), and use a table or point evaluation to see which choice has the same output as the original expression for that $x$ .

Step-by-step Explanation

Rewrite the fractional exponents as square roots

Notice that $x^{1/2} = \sqrt{x}$ and $x^{3/2} = (x^{1/2})^3 = (\sqrt{x})^3$ .

So the original expression

\frac{x^{3/2}-9x^{1/2}}{\sqrt{x}-3}

can be thought of in terms of powers of $\sqrt{x}$ .

Factor the numerator

Factor out the common factor $x^{1/2}$ (which is $\sqrt{x}$ ) from the numerator:

x^{3/2} - 9x^{1/2} = x^{1/2}(x - 9).

So the whole fraction becomes

\frac{x^{3/2}-9x^{1/2}}{\sqrt{x}-3} = \frac{x^{1/2}(x-9)}{\sqrt{x}-3}.

Use a difference of squares to factor $x-9$

Recognize that $x-9$ is a difference of squares:

$x = (\sqrt{x})^2$
$9 = 3^2$

x - 9 = (\sqrt{x})^2 - 3^2 = (\sqrt{x}-3)(\sqrt{x}+3).

Substitute this into the fraction:

\frac{x^{1/2}(x-9)}{\sqrt{x}-3} = \frac{\sqrt{x}(\sqrt{x}-3)(\sqrt{x}+3)}{\sqrt{x}-3}.

Since $x \ne 9$ , $\sqrt{x} \ne 3$ , so you can safely cancel the common factor $\sqrt{x}-3$ from numerator and denominator.

Simplify and match to an answer choice

After canceling $\sqrt{x}-3$ , you are left with

\sqrt{x}(\sqrt{x}+3).

Distribute $\sqrt{x}$ :

\sqrt{x}\cdot\sqrt{x} + 3\sqrt{x} = x + 3\sqrt{x}.

So the expression is equivalent to $x+3\sqrt{x}$ , which matches answer choice A.

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation