# Square Root of 260 — Value, Simplification, and Steps

TL;DR

The square root of 260 is \(2 \sqrt{65}\) in simplest radical form and approximately 16.125 as a decimal. This article shows how to simplify \(\sqrt{260}\) by prime factorization, estimate it between whole numbers, and explains why \(\sqrt{260}\) is an irrational number.

The square root of 78 is approximately **8.832** (irrational, so it never terminates or repeats). Because 78 has no square factor, \(\sqrt{78}\) cannot be simplified further.

> **Quick Answer:**
>
> **Result:** \(\sqrt{78} \approx 8.832\)
> 
> **Notation:** irrational decimal, non-terminating
> 
> **Method shown:** long division and estimation
> 
> **Approximate value:** 8.832 (to 3 decimal places)
> 
> **Exact form:** \(\sqrt{78}\), already in simplest radical form (78=2×3×13)

## Quick Reference Table

| Expression | Approx. Value | Perfect Square? |
| --- | --- | --- |
| \(\sqrt{64}\) | 8.000 | Yes |
| \(\sqrt{75}\) | 8.660 | No |
| \(\sqrt{78}\) | 8.832 | No |
| \(\sqrt{80}\) | 8.944 | No |
| \(\sqrt{81}\) | 9.000 | Yes |
| 782 | 6,084 | — |

## Where the Square Root of 78 Appears

\(\sqrt{78} \approx 8.832\) is the length of the diagonal of a box whose squared edge-lengths add to 78, for instance a rectangle with sides \(\sqrt{13}\) and \(\sqrt{65}\). It also appears as the distance between two points whose coordinate differences square to 78.

## What a Square Root Is

A **square root** of a number n is a value that, multiplied by itself, gives n. When n is a perfect square, the root is a whole number; when it is not, the root is irrational, a decimal that runs forever without repeating. Because 78 sits between the perfect squares 64 and 81, its square root falls between 8 and 9.

## Is the Square Root of 78 Rational or Irrational?

The square root of 78 is **irrational**. Its prime factorisation is 78=2×3×13, three distinct primes with no repeated pair, so no whole number squares to 78 and the decimal never terminates.

## How to Compute the Square Root of 78

### Method 1: Simplify the Radical (Check First)

List the prime factors of 78.

78=2×3×13

No prime appears twice, so there is no square factor to pull out.

**Final answer:** \(\sqrt{78}\) is already in simplest radical form.

### Method 2: Estimation

The nearest perfect squares are 64 and 81, so \(\sqrt{78}\) lies between 8 and 9.

**Final answer:** \(\sqrt{78} \approx 8.83\).

### Method 3: Long Division

Pair the digits from the right and add decimal pairs: 78.0000. The largest square ≤78 is 64; the next digits give 8.832…

**Final answer:** \(\sqrt{78} \approx 8.832\).

## Common Mistakes With Square Root of 260

### **Mistake 1: Trying to simplify a radical with no square factor**

**Where it slips in:** assuming every radical reduces to something smaller. **The correct way:** check the prime factors first.

### **Mistake 2: Rounding too early**

**Where it slips in:** writing \(\sqrt{78} = 8.8\) and treating it as exact. **The correct way:** keep the exact form through the algebra.

### **Mistake 3: Confusing 78 with a nearby perfect square**

**Where it slips in:** expecting a clean whole-number answer. **The correct way:** recognize that 78 is not a perfect square, so its root is irrational.

## Where to Go From Here

Estimate \(\sqrt{75}\) and \(\sqrt{80}\) by the same method, then check your guesses against the table above. To build these skills with a teacher, explore Bhanzu's online math classes.

## Frequently Asked Questions

### What is the square root of 260 in simplest radical form?

It is \(2 \sqrt{65}\).

### Is the square root of 260 rational or irrational?

Irrational. The root cannot be expressed as a fraction.

### What is \(\sqrt{260}\) as a decimal?

Approximately 16.125.

### Can \(\sqrt{260}\) be simplified further than \(2\sqrt{65}\)?

No. The remaining radicand 65 has no repeated prime factor, so it is fully simplified.