Math: 40c09d66

ID: 40c09d66

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

Comment: A tricky question on rules of exponents.

Method 1: Convert the expression on the left-hand side using fractional exponents. The power under the root stays the same and becomes the numerator; the number outside the radical becomes the denominator.

(x^(5/2))/(x^(4/3))

x^(5/2 - 4/3)

x^(15/6 - 8/6)

x^(7/6)

The correct answer is 7/6.

Method 2: A graph is not likely to prove too useful, but you could plug in a value for "x" and then look to get close to that value by using a slider, perhaps. How about x = 3? To input a cube root in Desmos, click on the keyboard icon in the bottom-left, then the "functions" button, then scroll down to the "NUMBER THEORY" sub-section.

So, according to the expression, the function should equal 3.60281086553 when x = 3. Now, create a second expression, 3^n (or whatever letter you like). Add a slider and aim to match the value above. With a little trial and error, you should be close enough to get the correct answer. For starters, the value jumps from about 3.348 when n = 1.1 to 3.737 when n = 1.2. Thus, it must be true that n is between these values, and you can readjust the range from 1.1 to 1.2, using a step of 0.001, since the SAT requires answers rounded to the thousandths, unless you are prompted otherwise.

As you can see, a value of 1.166 is slightly under the target number.

One bump further to the right, and 1.167 overshoots the target. Because 3.6041 is slightly closer to 3.6028 than 3.6001 is to 3.6028, I might favor 1.167. However, the SAT is pretty generous when it comes to rounding, and either 1.166 or 1.167 is considered a correct answer.

Method 3: With no answer choices to tinker with, this question is automatically more difficult, and one of the two methods above ought to be useful over other methods. There is no need to invoke calculus or anything: the SAT tests concepts that most students have seen in a regular geometry and algebra II curriculum.