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Equivalent expressions — SAT practice questions

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Practice 140 Equivalent expressions questions in the app

What this topic is

Equivalent expressions on the SAT Math test means rewriting an algebraic form so it matches another form that looks different but is identical for all allowed values of the variable.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Mid

Which expression is equivalent to $\dfrac{2}{x-3}+\dfrac{x}{x^2-9}$, where $x\ne3$ and $x\ne-3$?

A$\dfrac{3x+6}{x-3}$✗ Not this oneB$\dfrac{3}{x-3}$✗ Not this oneC$\dfrac{3x+6}{x^2-9}$✓ CorrectD$\dfrac{2x+6}{x^2-9}$✗ Not this one

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Answer: C — $\dfrac{3x+6}{x^2-9}$

Note $x^2-9=(x-3)(x+3)$. Rewrite the first fraction: $\dfrac{2(x+3)}{(x-3)(x+3)}+\dfrac{x}{(x-3)(x+3)}=\dfrac{2x+6+x}{(x-3)(x+3)}=\dfrac{3x+6}{x^2-9}$.

Question 2Easier

Which expression is equivalent to $(x+4)^2-(x-4)^2$?

A$8x$✗ Not this oneB$16x$✓ CorrectC$2x^2+32$✗ Not this oneD$x^2-16$✗ Not this one

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Answer: B — $16x$

Using the difference of squares gives $[(x+4)-(x-4)][(x+4)+(x-4)]=8(2x)=16x$.

Question 3Easier

For $x>0$, which expression is equivalent to $\sqrt[3]{x^5}\cdot x^{1/3}$?

A$x$✗ Not this oneB$x^{5/9}$✗ Not this oneC$x^{4/3}$✗ Not this oneD$x^2$✓ Correct

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Answer: D — $x^2$

Adding exponents gives $x^{5/3}x^{1/3}=x^{6/3}=x^2$.

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