Square Roots Formula

The Formula

\sqrt{a} = b \iff b^2 = a, \quad b \geq 0

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

Quick Example

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

Notation

\sqrt{\phantom{x}} is the radical symbol

What This Formula Means

The square root of a number a is the non-negative value b such that b \times b = a; it is the inverse of squaring and is written \sqrt{a}. For example, \sqrt{25} = 5 because 5^2 = 25.

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

Formal View

\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 \sqrt{144}.

Solution

  1. 1
    Recall that a square root asks for the positive number whose square equals the original number.
  2. 2
    Ask: what number multiplied by itself gives 144?
  3. 3
    Test: 12 \times 12 = 144. So \sqrt{144} = 12.

Answer

12
The square root of a number n is the value that, when multiplied by itself, produces n. Memorizing perfect squares (1, 4, 9, 16, 25, ..., 144) makes these computations fast.

Example 2

medium
A square has an area of 196 cm². What is the side length of the square?

Common Mistakes

  • Thinking \sqrt{a+b} = \sqrt{a} + \sqrt{b}
  • Forgetting \pm when solving x^2 = k
  • Confusing \sqrt{x^2} with x — the correct result is |x| because square root always returns the non-negative value

Why This Formula Matters

Essential for distance formulas (Pythagorean theorem), solving quadratic equations, and geometry. Square roots appear in physics (speed from kinetic energy), statistics (standard deviation), and engineering (signal processing).

Frequently Asked Questions

What is the Square Roots formula?

The square root of a number a is the non-negative value b such that b \times b = a; it is the inverse of squaring and is written \sqrt{a}. For example, \sqrt{25} = 5 because 5^2 = 25.

How do you use the Square Roots formula?

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

What do the symbols mean in the Square Roots formula?

\sqrt{\phantom{x}} is the radical symbol

Why is the Square Roots formula important in Math?

Essential for distance formulas (Pythagorean theorem), solving quadratic equations, and geometry. Square roots appear in physics (speed from kinetic energy), statistics (standard deviation), and engineering (signal processing).

What do students get wrong about Square Roots?

Not all square roots are nice integers (\sqrt{2} \approx 1.414\ldots).

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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