Math: 2c88af4d

ID: 2c88af4d

(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Advanced Math)

Comment: Knowing how to toggle back and forth between radical notation (with a root symbol, √) and exponential notation is a must for a top-level test-taker, as well as rules of exponents.

Method 1: A negative exponent provides directions, more or less, to kick the base across the line (the division bar) and make the exponent positive. Thus, x^-2 in the numerator will become x^2 in the denominator. Add the fractional exponents of the x's, and you are good to go.

x^2 * x^1/3

x^6/3 * x^1/3

x^7/3

The correct answer must have an equivalent form of x^7/3 in the denominator. A glance at the answer choices shows that only (D) does so:

A. x^2/3 X

B. x^2/3 X

C. x√x --> √(x * x^2) --> √x^3 --> x^3/2 X

D. x^2 * {cube root}x --> {cube root}(x * x^2 * x^2 * x^2) --> {cube root}(x^7) --> x^7/3 √

The correct answer is (D).

In all, the solution should take under a minute, even without checking the y's. However, for the fact-checker or the person who wants to take a few seconds to ensure that the answer is correct, it is fine to look at the y's as well. In the denominator of the original expression, y^-1 will be shifted up to the numerator as y^1, and y^1/2 * y^1, or y^1/2 * y^2/2, is y^3/2. All of answer choices (B), (C), and (D) have the correct numerator.

Method 2: Nothing beats knowing how to work with exponents, but you can alter the expression for Desmos and add a slider for an unknown that you could just label a to replace y, since Desmos graphs functions in y equals something. As long as you observe the given constraint that y > 1 in the original expression (a > 1 in the version below), you should be able to enter the answer choices directly until the two graphs align perfectly.

The vertical asymptotes at x = -1 or 1 indicate that x is not allowed to be these values anyway, just as the question tells you with x > 1. All you have to do now is type in the answer choices. To type in roots other than a square root, select the "Show Keypad" icon in the bottom-left corner, then hit the "functions" button, and finally scroll all the way down to the "NUMBER THEORY" portion, where you will see n outside of a root symbol.

Interestingly, Desmos chose to make the equivalent expression the same color as the original. That certainly makes the job easier. The correct answer must be (D).

Method 3: Any calculator that can work with exponents can also solve the expression if you just plug in numbers. If you keep track of the decimal answer, it is virtually impossible for multiple answers to produce the same value. For example, let x = 2 and y = 3.

Now, plug in the same values into each answer choice to check for equivalence.

Notice that the first and fifth entries match perfectly. As long as you know how to use the calculator, then, it is clear that (D) is the correct answer.