The Meaning of a Square Root

Which number multiplied by itself gives the value under the sign? The roots worth memorising, why squaring a root returns the original number, and how two roots combine into one.

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A square root asks one question: which number, multiplied by itself, gives the number under the sign? Once that is clear, the notation stops being mysterious — and knowing a handful of common roots by heart makes most problems immediate.

Concept The basic idea

A square root is the same thing as an exponent of one half. Both notations describe the identical operation.

, because .

A root sign with no small number written on it always means a square root — the 2 is understood.

Note The roots worth memorising
√4 = 2√9 = 3√16 = 4 √25 = 5√36 = 6√49 = 7 √64 = 8√81 = 9 each one reverses a square: 2² = 4, 3² = 9, and so on
These eight come up constantly. Recognising them on sight removes most of the arithmetic from root problems.
Rule 1 Squaring a root

Squaring undoes a square root, returning the original number. This works even when the root itself is not a whole number:

You never need to evaluate to use this. The two operations cancel each other directly.

Rule 2 Multiplying roots

Two roots multiplied together become a single root of the product. This often turns an awkward pair into a familiar value:

Neither nor is a whole number, yet their product is exactly 4. Combining first is often far easier than evaluating each root separately.

Example Applying both rules

Evaluate and .

The first squares a root, so the two operations cancel: the answer is 11.
For the second, combine under one root: .
, since .
⟹ 11 and 10

Neither calculation required a calculator, because each rule was applied before any root was evaluated.

Note Mistakes to avoid
Reading as 4.5 — a square root is not half the number.
Assuming ; the multiplication rule has no addition counterpart.
Evaluating as a decimal when squaring it would give exactly 7.
Forgetting that a bare root sign means a square root.
Confusing with — one reverses the other.
Summary
  1. A square root asks which number multiplied by itself gives the value under the sign.
  2. In exponent form, √a is the same as a^(1/2), and a bare root sign always means a square root.
  3. Knowing √4 through √81 by heart makes most problems immediate.
  4. Squaring a root returns the original number: (√a)² = a, even when √a is not whole.
  5. Roots multiply into a single root: √a × √b = √(ab), which often produces a familiar value.
283. Relations and Functions
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