4.7 SAT Math - Square Roots with Variables

Starting SAT Score 600 and above:

Sometimes the SAT testmakers will try to freak you out by putting variables into the square roots questions.  These will still be solved just like regular square roots; again, the numbers get uglier, but the rules never change. 

Example Problem

Try this one:

6.	4√ x + 3√ x =

Don't worry about the x's under the square root, just follow the rules.  The rules say if you have the same number under the square root symbol, you can add the coefficients, so the answer would be

Of course, you could also use ITOS to solve, picking a number for x and getting a numerical answer, and then seeing which one matches.

Example Problem 2

Here's another one:

15. If

3√ 7x + 1  =  9

then x=

For this problem, you still solve it like any other square root.  Undo the cube root by cubing both sides:

Then solve like regular algebra:

Example Problem 3

Another kind will look like this:

13. If 0 < a < 1, which of the following gives the correct ordering of √a, a, and a²?
(A) a < √a < a²
(B) a < a² < √a
(C) √a < a < a²
(D) √a < a² < a
(E) a² < a < √a

The easiest way to solve this kind is to use ITOS.  Simply pick a number for a, solve for the other terms, and order them least to greatest. 

Let's try

a = 1 _________ 3

, or .33.

The square root of

1 _________ 3

is

1 _________ √3

, or .57, and

a2 = 1 _________ 9

, or .11.

  So the order they should go in is a² < a < √a, answer choice (E).

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