Square Root of 4 — Why It Equals 2, Methods, Examples

TL;DR

The square root of 4 is exactly 2 — rational, an integer, no decimal tail. The principal (positive) square root is 2, though -2 also squares to 4. This article covers why ( \sqrt{4} = 2 ), three methods to verify it (prime factorization, repeated subtraction, long division), where ( \sqrt{4} ) shows up, and the slip that trips students at the boundary between ( \sqrt{4} ) and ±2.

Result: ( \sqrt{4} = 2 ) (principal square root); both 2 and -2 satisfy ( x^2 = 4 ).

Approximate value: No approximation needed — the answer is exact.

Exact form: 2.

Quick Reference Table — Small Perfect Squares

( \sqrt{4} ) is the second-smallest non-trivial perfect-square root, sitting between ( \sqrt{1} = 1 ) and ( \sqrt{9} = 3 ).

Where √4 lives

( \sqrt{4} ) shows up wherever a square has area 4 — its side is 2. The Pythagorean triple (3,4,5) uses 2 as one leg; the equation ( x^2 = 4 ) has solutions ( x = , \pm 2 ), which is what algebra students meet the first time they solve a quadratic. In physics, a 2-second free fall covers ( \frac{1}{2} g t^2 ), and on Earth, ( t = \sqrt{4} = 2 ) seconds is the time to fall about 20 meters.

What "square root of 4" means

The square root of a non-negative number ( n ) is the value ( x ) such that ( x^2=nx^2 = n ). For ( \sqrt{4} ), the positive ( x ) with ( x^2=4 ) — which is 2, because 2⋅2=4.

The symbol ( \sqrt{\phantom{x}} ) denotes the principal (non-negative) square root. The equation ( x^2=4 ) has two solutions, ( x=2 ) and ( x=-2 ), but the expression ( \sqrt{4} ) refers only to 2.

Is √4 rational or irrational?

( \sqrt{4} = 2 ) is rational. A number is a perfect square if and only if every prime in its factorization appears to an even power. ( 4 = 2^2 ) — the 2 has exponent 2, which is even, so 4 is a perfect square and its square root is the integer 2.

How to find √4 — three methods

Method 1 — Prime factorization (Quick)

Factor 4 into primes, then pair them up.

( 4=2⋅2=2^2 )

The pair of 2s leaves the radical as a single 2:

( \sqrt{4} = \sqrt{2^2} = 2 )

Final answer: ( \sqrt{4} = 2.

Method 2 — Repeated subtraction (Standard)

Run repeated subtraction on 4. Subtract consecutive odd numbers: 4−1=3, 3−3=0. After two subtractions, the remainder is zero — so ( \sqrt{4} = 2 ).

Final answer: ( \sqrt{4} = 2 ) (the principal root).

Method 3 — Long division (Stretch)

Run long division on 4.

Step 1: Largest integer with square ≤ 4 is 2. Subtract: 4−4=0. Bring down 0.

Step 2: Double 2, find d, then subtract. Each further step gives d=0, so ( \sqrt{4} = 2.

Final answer: ( \sqrt{4} = 2.

Where students lose the mark on √4

1. Confusing the square root expression with the quadratic equation.

Don't do this: ( \sqrt{4} ) and write ±2.

The correct way: ( \sqrt{4} = 2 ) (the principal root). The ± shows up only when you solve ( x^2=4 ).

2. Treating 4 as if it were not a perfect square.

Don't do this: reporting decimal forms.

The correct way: ( \sqrt{4} = 2.

3. Misreading ( \sqrt{4} ) as a division or fourth root.

Correct way: ( \sqrt{4} = 2.

Wrapping Up

• The square root of 4 is 2 — the principal root.
• ( \sqrt{4} ) is rational because 4 is a perfect square.
• The expression resolves to a single value; the equation has two solutions.
• All three methods confirm ( \sqrt{4} = 2 ) — prime factorization, repeated subtraction, long division.

A Practical Next Step

1. Show that 9 = 3 using prime factorization.
2. Solve ( x^2 = 4 ) and explain the two values.
3. Find ( \sqrt{16} ) using repeated subtraction.