"Which Value of x Makes √(1 − √(2 − √x)) Undefined as a Real Number?" — GMAT® Worked Solution
From Episode 40 of Real GMAT® Problems (The GMAT® Strategy Podcast). For the full strategy behind square roots problems on the GMAT®, read: GMAT® Square Roots: FOIL, Nested Radicals, and Why Zero Is Not Negative.
The Problem
Source: Official Guide for GMAT® Review, 11th Edition
For which of the following values of x is √(1 − √(2 − √x)) not defined as a real number?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
The Setup
We have three nested square roots. The outermost radical covers the whole expression. Inside it: 1 − √(2 − √x). Inside that: 2 − √x. At the bottom: √x.
A square root is undefined as a real number when the value underneath it is negative. Zero is fine — √0 = 0, which is a real number. A positive number is fine. The only case that fails is a negative.
So the goal is to find the answer choice that produces a negative value somewhere under one of the radicals.
The reliable way to do this is to test each answer choice from the inside out. Plug in the value for x, evaluate √x first, then the middle layer, then check whether the outermost radical would be taking the square root of a negative.
Step 1: Test (A) x = 1
Innermost: √1 = 1
Middle: 2 − 1 = 1, then √1 = 1
Outermost: 1 − 1 = 0, then √0 = 0
The outermost expression evaluates to √0, which equals 0. Zero is a real number. So x = 1 is defined.
Not the answer.
Step 2: Test (B) x = 2
Innermost: √2 ≈ 1.41
Middle: 2 − 1.41 = 0.59, then √0.59 ≈ 0.77
Outermost: 1 − 0.77 = 0.23, then √0.23 ≈ 0.48
Positive value throughout. Defined.
Not the answer.
Step 3: Test (C) x = 3
Innermost: √3 ≈ 1.73
Middle: 2 − 1.73 = 0.27, then √0.27 ≈ 0.52
Outermost: 1 − 0.52 = 0.48, then √0.48 ≈ 0.69
Positive throughout. Defined.
Not the answer.
Step 4: Test (D) x = 4
Innermost: √4 = 2
Middle: 2 − 2 = 0, then √0 = 0
Outermost: 1 − 0 = 1, then √1 = 1
The outermost expression evaluates to 1. Real number. Defined.
Not the answer.
Step 5: Test (E) x = 5
Innermost: √5 ≈ 2.24
Middle: 2 − 2.24 = −0.24
Stop right here.
The middle layer asks us to take the square root of −0.24. That is the square root of a negative number, which is not defined as a real number on the GMAT®.
So x = 5 makes the expression undefined.
The answer is (E).
Why This Problem Matters
About 40% of test takers miss this one — roughly four times the miss rate of the warm-up. The math is not the hard part. The definition is.
Look at where the wrong answers cluster.
A large share of test takers pick (D) x = 4. They plug in 4, get 2 − 2 = 0 in the middle layer, see √0, and conclude that √0 is "not a real number." It is. Zero is neither positive nor negative on the GMAT®, and √0 = 0, a perfectly valid real number.
About 13% pick (A) x = 1. Same reasoning trap. They plug in 1, work through to the outermost layer, get √0, and conclude that means the expression is undefined. Same mistake, different starting point.
Together, that's roughly 21% of test takers who believe √0 is not a real number. It is.
The correct answer relies on the other half of the rule: √(negative) IS undefined. When x = 5, the middle layer becomes 2 − √5, and √5 is greater than 2, so the subtraction produces a negative. You cannot take the square root of a negative on the GMAT®, so the whole expression fails to be a real number.
Two things to take away if you missed this:
The first is the definitional rule itself. Square root of zero equals zero, and zero is real. Square root of any negative number is not real. Writing that down once and reviewing it a few times tends to stick.
The second is the inside-out approach. Plugging in starting from the innermost layer is much harder to mess up than trying to evaluate the whole nested expression at once. It costs a few extra seconds and prevents most of the silly errors that show up on these problems.
Previous problem: Multiplying Conjugate Radical Pairs: (√2+1)(√2−1)(√3+1)(√3−1)
Next problem: Squaring a Sum of Nested Radicals: (√(9+√80) + √(9−√80))² — GMAT® Worked Solution
Back to the strategy article: GMAT® Square Roots: FOIL, Nested Radicals, and Why Zero Is Not Negative