Why Distributing Roots Over Sums Breaks Down: The Square Root Shortcut Trap

The split imitates a move that really is allowed. Over multiplication it works: $\sqrt{4 \times 9} = \sqrt{36} = 6$, and $\sqrt{4}\,\sqrt{9} = 2 \times 3 = 6$ as well. So the hand reaches for the same move over a plus sign:

$$\begin{aligned} \sqrt{9 + 16} &= \sqrt{9} + \sqrt{16} \\ &= 3 + 4 \\ &= 7 \end{aligned}$$

Now compute the left side without splitting: $9 + 16 = 25$, and $\sqrt{25} = 5$. One expression has produced two values, 7 and 5, so the step producing 7 was not legal. Either the shortcut is wrong or 7 equals 5.

Those are the sides of a right triangle: $\sqrt{3^2 + 4^2} = 5$. The shortcut answer, 7, is the distance along both legs instead of across.

A much-cited analysis of how science students and working scientists respond to a result that clashes with what they believe found seven typical reactions. Only one is changing your mind; the other six set the awkward result aside^[1]. So this post computes both sides in full.

The property that does hold is this one:

$$\sqrt{ab} = \sqrt{a}\,\sqrt{b}, \quad \text{provided } a \ge 0 \text{ and } b \ge 0.$$

The addition version fails. A function $f$ with $f(a + b) = f(a) + f(b)$ for every input is called additive; tripling is additive, since $3(a + b) = 3a + 3b$. The square root is not, which is what a textbook means by calling it non-linear. One counterexample kills a claim that begins "for all", and 5 against 7 is one.

In one study, algebra students judged whether a written step was allowed by the shape the symbols made on the page rather than by what the rule says. The authors suggest this is why some algebra errors are so stubborn^[2].

Hold the two moves side by side. $\sqrt{ab} \to \sqrt{a}\,\sqrt{b}$ and $\sqrt{a + b} \to \sqrt{a} + \sqrt{b}$ make the same picture: one root becomes two, and the sign between the letters survives the trip. The pictures match; the rules underneath do not. Matching on the picture is not carelessness. On that study's reading, it is a trained eye doing what it was trained to do^[2].

A second habit pushes the same way. Researchers have long noticed that after years of mathematics in which quantities scale in step, students come to expect every rule to behave that way. In one teaching experiment, lessons built to break that expectation in eighth-graders working on geometry succeeded for most of the class, though some then began avoiding the scaling rule even on problems where it was the right one^[3].

Now compare the two sides in general. Take any $a \ge 0$ and $b \ge 0$, and square the split version:

$$\left(\sqrt{a} + \sqrt{b}\right)^2 = a + 2\sqrt{a}\sqrt{b} + b = (a + b) + 2\sqrt{a}\sqrt{b}.$$

Meanwhile $\left(\sqrt{a + b}\right)^2 = a + b$. So the square of the split version exceeds the square of the unsplit one by exactly $2\sqrt{a}\sqrt{b}$, which is never negative. Both sides are themselves non-negative, and among non-negative numbers the larger square belongs to the larger number. Therefore

$$\sqrt{a} + \sqrt{b} \ \ge\ \sqrt{a + b}.$$

Equality needs $2\sqrt{a}\sqrt{b} = 0$, which happens only when at least one of $a$ and $b$ is zero. Take $a = 0$ and $b = 16$: both sides give 4. Make both numbers positive and the gap opens at once, since $\sqrt{4 + 9} = \sqrt{13} \approx 3.606$ while $\sqrt{4} + \sqrt{9} = 5$.

So the shortcut does not merely give a different number: for two positive numbers it always gives a bigger one. A too-big answer still looks plausible, which is what makes this error hard to catch.

Try $\sqrt{1 + 4}$ before reading on. Work out both sides, then compare.

$$\sqrt{1} + \sqrt{4} = 1 + 2 = 3, \qquad \sqrt{1 + 4} = \sqrt{5} \approx 2.236.$$

Different answers, and the shortcut is the larger one again. Now write one sentence naming the illegal step in $\sqrt{1 + 4} = \sqrt{1} + \sqrt{4}$.

That check is the portable part of this post. Whenever a function looks like it should split across a plus sign, put two small numbers in and compute both sides; one disagreement kills the rule. At $a = b = 1$, squaring fails, since $(1 + 1)^2 = 4$ but $1^2 + 1^2 = 2$. Logarithms and sines fall the same way.

This post sets a wrong line beside a correct one on purpose, and the evidence for that layout is real but mixed^[4][5]. In one experiment, 74 fourth- and fifth-graders learning about decimals compared a wrong answer with a correct one, and came out better at both the method and the idea behind it than classmates comparing two correct answers^[4]. A review that looked at 55 studies together, from primary school through university, is less flattering: students who worked through fully solved examples did better in mathematics on average, but studies using only correct examples tended to show bigger gains than those built around mistakes^[5].

Saying out loud why the wrong line is wrong sits in the same mixed position. In one study run in beginning algebra classes, students who explained worked-out answers while they practised understood the ideas better than students who only practised^[6]. The same review found the opposite pattern: studies that added prompts to explain the examples showed smaller gains, not bigger ones^[5]. Treat the one-sentence explanation as reasonable to try, not as settled.

Order may matter too. In two experiments teaching university-level learners to solve probability problems, mixing wrong solutions in with correct ones helped those who already knew something about the topic handle harder, unfamiliar problems, while learners who knew very little did better with correct solutions only^[7]. The decimals experiment found no such split^[4]. My own judgment, given that disagreement: if $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$ is not yet solid for you, make it solid first.

None of these studies is about square roots, or radicals, or algebraic notation at all, beyond the one study of how algebra students read symbol shapes^[2]. The idea that the shape of $\sqrt{ab}$ drags people into splitting $\sqrt{a + b}$ applies that study's mechanism to a case it never looked at^[2].

The people studied were not you, and the distance matters: fourth- and fifth-graders on decimals^[4], eighth-graders on geometry^[3], university-level learners on probability^[7], beginning algebra classes^[6], science students meeting results that contradicted them^[1]. The review is the broadest, spanning primary school through university, but it covered solved examples in general, not this error^[5]. No study here measured whether writing this shortcut once, and being shown 5 against 7, stops a person writing it again.

None of that touches the mathematics, which is proved above. What is thin is the advice on how to practise.

One last caution from that geometry experiment: some students broke the habit so thoroughly that they stopped using the scaling rule where it genuinely applied^[3]. Do not walk away deciding that roots never split. They split over multiplication, where $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$ for non-negative $a$ and $b$; they do not split over addition. Losing the valid rule would set you back further than the invalid one ever did.