"√(16 × 20 + 8 × 32) = ?" — GMAT® Worked Solution

Source: Official Guide for GMAT® Review, 11th Edition

$\sqrt{16 \times 20 + 8 \times 32} = ?$

(A) $4\sqrt{20}$

(B) 24

(C) 25

(D) $4\sqrt{20} + 8\sqrt{2}$

(E) 32

Try it before reading on.

Route 1: Brute Force and a Memorized Square

Nothing wrong with running the arithmetic on this one, following the order of operations from the inside out:

$16 \times 20 = 320$

$8 \times 32 = 256$

$320 + 256 = 576$

That leaves $\sqrt{576}$. If the perfect squares are memorized this far, 576 registers on sight as $24^2$.

The answer is (B).

Plenty of test takers stall at exactly that step, and it's an understandable place to stall. The next two routes finish the problem without leaning on memorization at all.

Route 2: Prime Factor the Way Through

Prime factoring means dividing by prime numbers until no more prime results come out. Two is the place to start: it's the smallest prime, and the only even prime, which is worth knowing because the exam likes that trap.

576 ÷ 2  = 288
288 ÷ 2  = 144
144 ÷ 2  =  72
 72 ÷ 2  =  36
 36 ÷ 2  =  18
 18 ÷ 2  =   9
  9 ÷ 3  =   3

Six 2s and two 3s, and every integer greater than 1 has exactly one prime factorization, its fingerprint:

$576 = 2^6 \times 3^2$

The move from there is treating the square root as an exponent of $\frac{1}{2}$, which cuts every exponent in half:

$576^{1/2} = 2^3 \times 3^1 = 8 \times 3 = 24$

The same (B) again, this time without needing $24^2$ in memory.

Route 3: Factor Out the Perfect Square

This is the more advanced route, and it takes some x-ray vision into the numbers. It starts from the fact that roots distribute over multiplication but not over addition, which is what makes $\sqrt{16 \times 20 + 8 \times 32}$ awkward as written. Factoring fixes the awkwardness, because factoring turns addition into multiplication.

A perfect square is already sitting in the first term: 16 in $16 \times 20$. The second term hides a perfect square one level down: 32 factors as $2 \times 16$, so $8 \times 32$ rewrites as $8 \times 2 \times 16$:

$\sqrt{16 \times 20 + 16 \times 16}$

Now 16 factors out of both terms:

$\sqrt{16(20 + 16)} = \sqrt{16 \times 36}$

Two perfect squares under one root, and roots distribute over multiplication, so each one comes out individually:

$\sqrt{16} \times \sqrt{36} = 4 \times 6 = 24$

Why This Problem Matters

The memorization baseline here is perfect squares through $20^2$, with an optional extension to $30^2$ for anyone wanting a cushion. This problem is the exception that reaches past 20, since 576 is $24^2$, and even people who memorized that far sometimes don't notice in the moment. Seeing all three routes means no single gap should stall the question: memory slips, and prime factoring finishes; the factoring insight doesn't arrive, and brute force finishes anyway.

Which route to favor is a personal call, and any of the three can work; what matters is which one a person trusts under pressure. The one rule worth keeping is that the best route is the safest route, meaning the one least likely to cause a mistake. The biggest score killer in the quant section probably isn't lack of speed; it's missing questions you know how to do. Survey the approaches, pick the one that holds up under pressure, and let speed come from repetition instead of rushing.

Want the full strategy? Read: GMAT® Quant Solution Paths: Theory, Plugging In, and Brute Force

From Episode 9 of Real GMAT® Problems (The GMAT® Strategy Podcast).