"(√(9+√80) + √(9−√80))²" — 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: Retired GMAT® practice test problem

(√(9 + √80) + √(9 − √80))² =

(A) 1

(B) 9 − 4√5

(C) 18 − 4√5

(D) 18

(E) 20

The Setup

We need to square the sum of two nested radicals. The cleanest approach is to write out the square as a product — the expression times itself — and FOIL the whole thing.

Let:

A = √(9 + √80)

B = √(9 − √80)

We need (A + B)².

That is the same as (A + B)(A + B), which we can FOIL directly.

Step 1: Write the Square as a Product and FOIL

(A + B)(A + B)

First: A × A = (√(9 + √80))² = 9 + √80

(The outer square root and the squaring cancel each other out, leaving what was inside.)

Outside: A × B = √(9 + √80) × √(9 − √80)

Inside: B × A = √(9 − √80) × √(9 + √80)

Last: B × B = (√(9 − √80))² = 9 − √80

Step 2: Combine the First and Last Terms

First + Last:

(9 + √80) + (9 − √80) = 18

The √80 terms cancel. What's left is 18.

So far: 18 + (Outside) + (Inside)

Step 3: Combine the Middle Terms

The Outside and Inside terms are the same — multiplication can happen in either order.

Outside = Inside = √(9 + √80) × √(9 − √80)

So the middle becomes:

2 × √(9 + √80) × √(9 − √80)

Now use the rule √a × √b = √(ab). The two radicals combine under a single root:

2 × √((9 + √80)(9 − √80))

Step 4: FOIL Inside the Radical

We have another conjugate pair: (9 + √80)(9 − √80).

First: 9 × 9 = 81

Outside: 9 × (−√80) = −9√80

Inside: √80 × 9 = 9√80

Last: √80 × (−√80) = −80

Combining:

81 − 9√80 + 9√80 − 80

Middle terms cancel:

81 − 80 = 1

Step 5: Put It Back Together

The middle of our expression becomes:

2 × √1 = 2 × 1 = 2

Combining with the 18 from Step 2:

18 + 2 = 20

The answer is (E).

Why This Problem Matters

About 31% of test takers miss this one — about three times the miss rate of the warm-up. By the numbers, it is one of the harder problems in the Official Guide on this topic.

Look at what we actually did, though.

FOIL on a sum (Steps 1–3). The rule √a × √b = √(ab) to combine two radicals into one (Step 3). FOIL again on a conjugate pair to simplify what's inside (Step 4). That is the same set of skills as the warm-up problem — applied twice instead of once, with one extra combination step in the middle.

A lot of harder GMAT® quant problems are built this way. They are not testing a new skill. They are testing whether the same basic skills can be applied three or four times in a row without losing the thread or making a sign error.

Three things to take away:

Recognize the conjugate pattern. Twice in this problem, (a + b)(a − b) collapsed because the middle terms canceled. Once you see that pattern, expressions that look intimidating become routine.

Take the leap of faith on messy setups. After Step 2, the expression still looked rough — there were nested radicals inside other radicals. The cleanup did not become obvious until we kept going. If the FOIL is set up correctly, the simplification often only reveals itself after one or two more steps.

Write every FOIL term. Both times we used FOIL in this problem, the magic happened when the middle terms canceled. That cancellation is impossible to see if half the work is in your head and only the combined result makes it to the page.

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