# 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

| n   | \( \sqrt{n} \) (exact) | Perfect square? | Rational / irrational |
| --- | ---                     | ---              | ---                   |
| 1   | 1                       | Yes              | Rational              |
| 2   | \( \, \sqrt{2} \)         | No               | Irrational            |
| 3   | \( \, \sqrt{3} \)         | No               | Irrational            |
| **4** | **2**                 | **Yes**          | **Rational**          |
| 5   | \( \, \sqrt{5} \)         | No               | Irrational            |
| 9   | 3                       | Yes              | Rational              |
| 16  | 4                       | Yes              | Rational              |
| 25  | 5                       | Yes              | Rational              |
| 36  | 6                       | Yes              | Rational              |
| 49  | 7                       | Yes              | Rational              |

\( \sqrt{4} \) 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.