Square Roots Formula

Square roots are the non-negative number b such that b^2 = a, written sqrt(a) = b — the inverse of squaring.

The Formula

x=n   means   n2=x\sqrt{x}=n\;\text{ means }\;n^2=x

When to use: 25\sqrt{25} asks: what number times itself equals 25? Answer: 5.

Quick Example

16=4\sqrt{16} = 4 because 4×4=164 \times 4 = 16; and 2≈1.414\sqrt{2} \approx 1.414 (irrational).

Notation

The principal square root is the nonnegative number whose square is the radicand.

What This Formula Means

The non-negative number bb such that b2=ab^2 = a, written a=b\sqrt{a} = b — the inverse of squaring.

25\sqrt{25} asks: what number times itself equals 25? Answer: 5.

Formal View

∀a≥0:a=b  ⟺  b≥0∧b2=a. Equivalently, a=a1/2\forall a \geq 0: \sqrt{a} = b \iff b \geq 0 \land b^2 = a. \text{ Equivalently, } \sqrt{a} = a^{1/2}

Worked Examples

Example 1

easy
Find 144\sqrt{144}.

Answer

1212

First step

1
Recall that a square root asks for the positive number whose square equals the original number.

Full solution

  1. 2
    Ask: what number multiplied by itself gives 144?
  2. 3
    Test: 12×12=14412 \times 12 = 144. So 144=12\sqrt{144} = 12.
The square root of a number nn is the value that, when multiplied by itself, produces nn. Memorizing perfect squares (1, 4, 9, 16, 25, ..., 144) makes these computations fast.

Example 2

medium
A square has an area of 196196 cm². What is the side length of the square?

Example 3

medium
A right triangle has legs 66 and 88. Find the hypotenuse.

Common Mistakes

  • Dividing by 2 instead of finding a self-product — 36=6\sqrt{36}=6, not 18.
  • Forgetting non-perfect squares need estimates or radical form — 2\sqrt{2} is not a tidy decimal.
  • Ignoring the principal root convention — 36\sqrt{36} means 6, even though (−6)2=36(-6)^2=36.

Why This Formula Matters

Square roots connect exponents, area, the Pythagorean theorem, distance, and irrational numbers. Recognizing roots as inverse squares prevents calculator-only thinking. Recognizing it by "What number multiplied by itself gives the radicand?" — rather than by familiar numbers — is what lets a student tell it apart from cube roots and exponents in a mixed problem set.

Frequently Asked Questions

What is the Square Roots formula?

The non-negative number bb such that b2=ab^2 = a, written a=b\sqrt{a} = b — the inverse of squaring.

How do you use the Square Roots formula?

25\sqrt{25} asks: what number times itself equals 25? Answer: 5.

What do the symbols mean in the Square Roots formula?

The principal square root is the nonnegative number whose square is the radicand.

Why is the Square Roots formula important in Math?

Square roots connect exponents, area, the Pythagorean theorem, distance, and irrational numbers. Recognizing roots as inverse squares prevents calculator-only thinking. Recognizing it by "What number multiplied by itself gives the radicand?" — rather than by familiar numbers — is what lets a student tell it apart from cube roots and exponents in a mixed problem set.

What do students get wrong about Square Roots?

The procedure for square roots is the easy part; the trap is dividing by 2 instead of finding a self-product. Asking "What number multiplied by itself gives the radicand?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Square Roots formula?

Before studying the Square Roots formula, you should understand: exponents, multiplication.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Cube Roots, Square Roots, and Irrational Numbers →
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