# Square Root of 100 — Value, Method & Examples

## TL;DR

The square root of 100 is exactly **10**, because 10×10=100 — so 100 is a perfect square with a clean whole-number root. This article gives the value, three ways to compute it, where \( \sqrt{100} \) shows up, and the slip-ups students make most often.

## The Square Root of 100 is 10

The square root of 100 is **10**. It is a perfect square: 10×10=100, so \( \sqrt{100} = 10 \) — no decimal, no rounding.

It is also the round-number anchor of the square-root table from 1 to 100: \( \sqrt{100} \) closes the run of perfect squares that starts at 1.

> ### Quick Answer:
> **Result:** \( \sqrt{100} = 10 \)
> **Notation:** radical form \( \sqrt{100} \); exponent form \( 100^{1/2} \).
> **Method shown:** prime factorisation, with cross-checks by long division and repeated subtraction.
> **Rational or irrational:** rational — \( 10 = \frac{10}{1} \).
> **Exact form:** 10 (an integer; the radical fully resolves).

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

| n   | \( \sqrt{n} \) | Perfect square? |
| --- | -------------- | ---------------- |
| 49  | 7              | yes              |
| 64  | 8              | yes              |
| 81  | 9              | yes              |
| **100** | **10**       | **yes**         |
| 121 | 11             | yes              |
| 144 | 12             | yes              |
| 169 | 13             | yes              |
| 196 | 14             | yes              |
| 225 | 15             | yes              |
| 256 | 16             | yes              |

These are the perfect squares bracketing 100, with whole-number roots on either side — useful for estimating any nearby non-perfect square at a glance.

## Where The Square Root of 100 Appears

The result \( \sqrt{100} = 10 \) is everywhere the decimal system is, because 100=10² is the base of our place value — hundreds, percentages, and centi-units all rest on it. Geometrically, a square of area 100 square units has a side of \( \sqrt{100} = 10 \) units. It also turns up in the Pythagorean theorem: a 6–8–10 right triangle has a hypotenuse of 10, which is just the 3–4–5 triangle scaled by two.

## What "square root of 100" Means

A **square root** of a number \( n \) is a value \( x \) for which \( x^2 = n \). For 100, that value is 10, since \( 10^2 = 100 \).

The radical symbol \( \sqrt{;}; \) returns the **principal** (non-negative) root, so \( \sqrt{100} = 10 \). The equation \( x^2 = 100 \) has two solutions, 10 and −10, but the symbol \( \sqrt{100} \) alone names only the positive one.

## How To Find The Square Root of 100

### Method 1 — Prime factorisation

Factor 100 into primes, pair them, and take one from each pair.

\( 100 = 2 \times 2 \times 5 \times 5 = (2 \times 2)(5 \times 5) \)

One 2 and one 5 leave their pairs: \( \sqrt{100} = 2 \times 5 = 10 \).

**Final answer:** \( \sqrt{100} = 10 \).

### Method 2 — Long division

Pair the digits from the right: \( 1,00 \). The largest integer whose square is at most 1 is 1, so the first quotient digit is 1, with remainder 0. Bring down 00 to get a dividend of 00; double the quotient (1 → 2) and find \( d \) with \( (20+d) \cdot d \le 0 \), which gives \( d=0 \). The quotient is 10 with remainder 0, so \( \sqrt{100} = 10 \) exactly.

**Final answer:** \( \sqrt{100} = 10 \).

### Method 3 — Repeated subtraction of odd numbers

Subtract successive odd numbers and count the steps to reach 0: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.

That is 10 odd numbers, and 1+3+⋯+19=100, so \( \sqrt{100} = 10 \).

## Examples of Square Root of 100

### Example 1

**Evaluate \( \sqrt{100} \) directly.**

\( 10^2 = 100 \), so \( \sqrt{100} = 10 \). A whole number — no estimation needed.

### Example 2

A student is asked to simplify \( 100 + \sqrt{100} + \sqrt{100} \).

_Wrong attempt._ Reading it as "the root of 100 + 100 + 100," the student writes \( 200 \approx 14.14 \).

_Correct._ The two radicals are separate terms: \( 100 + \sqrt{100} + \sqrt{100} = 10 + 10 = 20 \). Square roots are evaluated before adding — the expression is \( 10 + 10 \), not \( 200 \).

### Example 3

**Simplify \( 100x^2 \) for x>0.**

\( \sqrt{100x^2} = \sqrt{100} \cdot \sqrt{x^2} = 10x \). The root distributes over a product.

### Example 4

**Find the side of a square whose area is 100 cm².**

\( \text{side} = \sqrt{100} = 10 	ext{ cm} \). Area = side², so the side is the square root of the area.

### Example 5

**A right triangle has legs 6 and 8. Find the hypotenuse.**

\( c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \). The 6–8–10 triangle.

## Common Mistakes With the Square Root of 100

### Mistake 1: Confusing \( \sqrt{100} \) with 100 ÷ 2

**Where it slips in:** A learner mistakenly reaches for division instead of asking what value squares to 100.

**Don't do this:** \( \sqrt{100} = 50 \).

**The correct way:** \( \sqrt{100} \) asks "what number times itself gives 100?" — that's 10.

### Mistake 2: Writing \( \sqrt{100} \) as ±10

**Where it slips in:** Confusing the radical with the solutions of \( x^2=100 \).

**Don't do this:** \( \sqrt{100} = ±10 \).

**The correct way:** \( \sqrt{100} = 10 \). The radical returns only the principal root.

### Mistake 3: Splitting the root over a sum

**Where it slips in:** Treating \( 36+64 \) as \( \sqrt{36} + \sqrt{64} \).

**Don't do this:** \( 36+64=6+8=14 \).

**The correct way:** \( 36+64=100=\sqrt{100} = 10 \). Square roots do not distribute over addition.

## Conclusion

- The **square root of 100** is 10, an exact whole number, because 100 is a perfect square.
- Prime factorisation gives it directly: \( 100 = 2^2 \times 5^2 \), so \( \sqrt{100} = 2 \times 5 = 10 \).
- \( \sqrt{100} \) names only the principal root; the equation \( x^2=100 \) is what yields ±10.
- Because 10 is an integer, \( \sqrt{100} \) is rational.