# Square Root of 105 — Value and How to Find It

## TL;DR

The square root of 105 is \( \sqrt{105} \approx 10.247 \), an irrational number that cannot be simplified because 105 = 3 × 5 × 7 has no repeated prime factor. This article shows the factorization check, the long-division method, where it appears, common mistakes, and worked examples.

#### Last updated on July 20, 2026

The square root of 105 is \( \sqrt{105} \approx 10.247 \), and it stays under the radical because 105 has no square factor.

> **Quick Answer:**  
> **Result:** \( \sqrt{105} \approx 10.247 \)  
> **Notation:** \( \sqrt{105} \) or \( 105^{1/2} \)  
> **Method shown:** long division and prime-factorization check  
> **Approximate value:** 10.247 (to 3 decimal places, irrational)  
> **Exact form:** \( \sqrt{105} \) (already in simplest radical form)

## Quick Reference Table

| Number | Simplified square root | Decimal (3 dp) |
| --- | --- | --- |
| \( 100 \) | \( 10 \) | \( 10.000 \) |
| \( 104 \) | \( 2\sqrt{26} \) | \( 10.198 \) |
| \( 105 \) | \( \sqrt{105} \) | \( 10.247 \) |
| \( 106 \) | \( \sqrt{106} \) | \( 10.296 \) |
| \( 108 \) | \( 6\sqrt{3} \) | \( 10.392 \) |
| \( 121 \) | \( 11 \) | \( 11.000 \) |

## Where the Square Root of 105 Appears

The square root of 105 is the side length of a square whose area is 105 square units. It also shows up as a diagonal: a rectangle with sides measuring roughly \( 5	ext{ and }21 \) has a diagonal of 5 + 100 style calculations, and any right triangle whose legs square-sum to 105 has a hypotenuse of exactly \( \sqrt{105} \).

## What Is the Square Root of 105?

The square root of a number is the value that, multiplied by itself, gives that number. Since 105 is not a perfect square, it sits between \( 10^2 \) and \( 11^2 \), so its square root is between 10 and 11.

The result is an irrational number that never terminates and never repeats. Its prime factorization, 105 = 3 × 5 × 7, has no repeated factor, so nothing can leave the radical.

## How to Find the Square Root of 105 (Methods)

### Method 1: Prime-factorization check

Break 105 into prime factors.

105=3×5×7

A factor leaves a square root only when it appears twice. Every prime here appears once.

\( \sqrt{105} = \sqrt{3 \times 5 \times 7} \)

**Final answer:** \( \sqrt{105} \) does not simplify; it stays as \( \sqrt{105} \).

### Method 2: Long division for the decimal value

Pair the digits of 105 from the right: 1|051. The largest square below 1 is 1, so the first digit is 1.

Bring down 05 to get 5, and find a digit \( x \) with \( 2x \times x \leq 500 \). Continuing the process gives the next decimals.

\( 105 \approx 10.247 \)

**Final answer:** \( 105 \approx 10.247 \) to three decimal places.

## Common Mistakes With Square Root of 105

### Mistake 1: Trying to pull a factor out

**Correct way:** Check the factorization first. Since 105 has no repeated prime, \( \sqrt{105} \) is already simplest.

### Mistake 2: Rounding too early

**Correct way:** Carry the division to the required precision, \( \approx 10.247 \) before rounding.

### Mistake 3: Confusing the square of 105 with its square root

**Correct way:** The square root asks for the number whose square is 105, which is \( \approx 10.247 \).

## Frequently Asked Questions

**What is the value of the square root of 105?**  
It is \( \sqrt{105} \approx 10.247 \), an irrational number.

**Is the square root of 105 rational or irrational?**  
Irrational. Since 105 is not a perfect square, it is a non-terminating, non-repeating decimal.

**Can \( \sqrt{105} \) be simplified?**  
No. Since 105 has no repeated prime factor, no whole number can leave the radical.

**Is 105 a perfect square?**  
No. It lies between the perfect squares 100 and 121.

**What is the negative square root of 105?**  
It is \( -\sqrt{105} \approx -10.247 \), since both \( +10.247 \) and \( -10.247 \) square to about 105.