"(√2+1)(√2−1)(√3+1)(√3−1)" — 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

(√2 + 1)(√2 − 1)(√3 + 1)(√3 − 1) =

(A) 2

(B) 3

(C) 2√6

(D) 5

(E) 6

The Setup

We have four parenthesized expressions multiplied together. Two contain √2. Two contain √3.

The cleanest path is to FOIL the √2 pair, FOIL the √3 pair, and multiply the two results.

FOIL stands for first, outside, inside, last. It's the standard way to expand the product of two binomials. If it has been a while since you used it, the Math Basics episode on quadratics covers the mechanics in detail.

Step 1: FOIL the √2 Pair

(√2 + 1)(√2 − 1)

First: √2 × √2 = 2

Outside: √2 × (−1) = −√2

Inside: 1 × √2 = √2

Last: 1 × (−1) = −1

Combining:

2 − √2 + √2 − 1

The middle terms cancel: −√2 + √2 = 0.

What's left: 2 − 1 = 1

Step 2: FOIL the √3 Pair

(√3 + 1)(√3 − 1)

First: √3 × √3 = 3

Outside: √3 × (−1) = −√3

Inside: 1 × √3 = √3

Last: 1 × (−1) = −1

Combining:

3 − √3 + √3 − 1

The middle terms cancel again.

What's left: 3 − 1 = 2

Step 3: Multiply the Two Results

(1)(2) = 2

The answer is (A).

Why This Problem Matters

This is the warm-up of the set. About 10% of test takers miss it, which is on the low end. The math itself is not complex — but the problem introduces a pattern that shows up over and over on the GMAT®, in this category and others.

The pattern is the conjugate pair: (√a + b)(√a − b).

When you FOIL a conjugate pair, the middle terms always cancel out. What is left is a − b² (the first term squared minus the last term squared). On this problem, that means each pair simplifies to a clean integer with no radicals left, which makes the final multiplication trivial.

If you recognize the pattern, you may be able to skip the full FOIL and write 2 − 1 = 1 for the √2 pair and 3 − 1 = 2 for the √3 pair directly. That works once the habit is built.

Until then, write every step.

The most common way to miss this problem is dropping a negative sign or forgetting a middle term while doing the FOIL in your head. Writing all four terms — first, outside, inside, last — on separate lines almost eliminates that risk. The few extra seconds it takes pay off many times over once the problems get harder.

This same FOIL setup, applied with one extra step, handles the much tougher third problem in this episode. Same skill. More layers. Worth getting comfortable here first.

Next problem: Which Value of x Makes the Nested Square Root Undefined? — GMAT® Worked Solution

Back to the strategy article: GMAT® Square Roots: FOIL, Nested Radicals, and Why Zero Is Not Negative