# Square Root of 25 — Value, Method, and Examples

TL;DR

The square root of 25 is exactly 5, because 5×5=25 — making 25 a perfect square with a whole-number root. This article gives the value, three ways to find it, where 25 turns up, and why the radical answer is 5 rather than ±5.

## The Square Root of 25 Is 5

The square root of 25 is **5**. It is a perfect square: 5×5=25, so 25=5 with no decimal tail and no rounding.

This is one of the first square roots worth knowing by heart — it anchors the 3–4–5 triangle, the most-used right triangle in school geometry.

## Quick Answer

### **Quick Answer:**

**Result:** 25=5  
**Notation:** radical form 25; exponent form 25^{1/2}.  
**Method shown:** prime factorisation, with cross-checks by repeated subtraction and long division.  
**Rational or irrational:** rational — 5 can be written as \tfrac{5}{1}.  
**Exact form:** 5 (an integer; no radical remains).

## Quick Reference Table — Square Roots of Nearby Perfect Squares

| n   | n√n | Perfect square? |
| --- | --- | --- |
| 1   | 1   | yes |
| 4   | 2   | yes |
| 9   | 3   | yes |
| 16  | 4   | yes |
| **25** | **5** | **yes** |
| 36  | 6   | yes |
| 49  | 7   | yes |
| 64  | 8   | yes |
| 81  | 9   | yes |
| 100 | 10  | yes |

Every value in this table is a whole number — these ten are the perfect squares from 1 to 100, and knowing them on sight makes most square-root work fast.

## Where the square root of 25 appears

The 25 result sits at the center of the 3–4–5 right triangle: legs of 3 and 4 give a hypotenuse of \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. That single triple shows up in the Pythagorean theorem, in carpentry for squaring a corner, and in the distance between the points (0,0) and (3,4). It also appears whenever a quadratic has a discriminant of 25, since 25 keeps the roots rational.

## What "square root of 25" Means

A **square root** of a number n is a value x for which x^2 = n. For 25, the value is 5, since 5^2 = 25.

The radical symbol √ asks for the **principal** (non-negative) root, so 25=5. The equation x^2=25 has two solutions, x=5 and x=−5, but the symbol 25 alone names only the positive one.

## How To Find The Square Root of 25

### Method 1 — Prime factorisation

Break 25 into prime factors, then pair them.

25=5×5  
One factor leaves each pair, so 25=5. This is the cleanest method for any perfect square.

**Final answer:** 25=5.

### Method 2 — Repeated subtraction of odd numbers

Subtract successive odd numbers from 25 and count the steps until you reach 0.

25−1=24,  24−3=21,  21−5=16,  16−7=9,  9−9=0.

Five subtractions reached zero, so 25=5. (This works because n^2 is the sum of the first n odd numbers.)

### Method 3 — Long division

Pair the digits: 25. The largest integer whose square is at most 25 is 5, and 5^2 = 25 exactly. Subtracting leaves a remainder of 0, so the root terminates.

**Final answer:** 25=5.

## Examples of Square Root of 25

### Example 1

**Evaluate 25 directly.**

5^2 = 25, so 25=5. A clean integer — no approximation needed.

### Example 2

A student is asked: does x^2=25 mean x=25?

_Wrong attempt._ Writing x=25 as the only answer drops a solution. Check: (−5)^2=25 as well, so −5 also satisfies the equation.

_Correct._ The **equation** x^2=25 has two roots, x=±5. The **symbol** 25 names only the principal root, 5. The two ideas are different — solving an equation is not the same as evaluating a radical.

### Example 3

**Simplify 3√25.**

25=5, so 3×5=15.

### Example 4

**Find the hypotenuse of a right triangle with legs 3 and 4.**

c=\sqrt{3^2 + 4^2} = \sqrt{25} = 5.

### Example 5

**Solve 25x^2 for x>0.**

25x^2=5x. The square root distributes over a product (unlike over a sum).

## Where Students Trip Up on the Square Root of 25

### Mistake 1: Writing √25 as ±5

**Where it slips in:** A student confuses solving x^2=25 with evaluating the radical 25.

**Don't do this:** 25=±5.

**The correct way:** 25=5. The radical returns only the principal (positive) root; the ± belongs to equation-solving, not to the symbol itself.

### Mistake 2: Confusing the square root with halving

**Where it slips in:** Treating 25 as 25÷2.

**Don't do this:** 25=12.5.

**The correct way:** 25 asks "what times itself gives 25?" — the answer is 5, not half of 25.

### Mistake 3: Calling √25 irrational

**Where it slips in:** Lumping every square root in with other non-terminating square roots.

**Don't do this:** Assuming 25 never terminates.

**The correct way:** 25 is a perfect square, so 25=5 is a whole number — clearly rational.

## Conclusion

- The **square root of 25** is 5, an exact whole number, because 25 is a perfect square (5×5=25).

- 25 names only the principal root, 5; the equation x^2=25 is what gives ±5.

- Prime factorisation is the fastest route: 25=5×5, so 25=5.

- Because 5 is an integer, 25 is rational, not irrational.

- 25=5 is the heart of the 3–4–5 right triangle.

## Frequently Asked Questions

Is the square root of 25 rational or irrational?

Rational. 25=5, and 5 can be written as the fraction \tfrac{5}{1}.

What is the square root of 25 in radical form?

It doesn't stay in radical form — 25 simplifies to the integer 5 because 25 is a perfect square.

What is −25?

There is no real square root of a negative number.