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# Square Root of 78 — Value, Simplification, and Steps

Algebra

TL;DR

The square root of 78 is irrational and approximately 8.832, and √78 is already in simplest radical form because 78 has no perfect-square factor. This article shows the estimation and long-division methods, gives a quick-reference table, and points to where √78 appears.

**Last updated on:** July 18, 2026 5 min read

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

> **Quick Answer:**  
> **Result:** √78 ≈ 8.832  
> **Notation:** irrational decimal, non-terminating  
> **Method shown:** long division and estimation  
> **Approximate value:** 8.832 (to 3 decimal places)  
> **Exact form:** √78, already in simplest radical form (78=2×3×13, no square factor)

## Quick Reference Table

| Expression        | Approx. Value | Perfect Square? |
|-------------------|---------------|------------------|
| √64               | 8.000         | Yes (8²)         |
| √75               | 8.660         | No (=5√3)       |
| √78               | 8.832         | No (simplest form) |
| √80               | 8.944         | No (=4√5)       |
| √81               | 9.000         | Yes (9²)         |
| 782               | 6,084         | —                |

## Where the Square Root of 78 Appears

√78 is the length of the diagonal of a box whose squared edge-lengths add to 78, for instance, a rectangle with sides √13 and √65, since 13 + 65 = 78. It also appears as the distance \(√{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) between two points whose coordinate differences square to 78, such as the points (0,0) and (7,√29).

## 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=8² and 81=9², 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. That same factorisation is why √78 is already in simplest radical form.

## 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:** √78 is already in simplest radical form.

### Method 2: Estimation

The nearest perfect squares are 64=8² and 81=9², so √78 lies between 8 and 9.

Since 78 is much closer to 81 than to 64, the root is close to 9.

Test 8.82=77.44 and 8.92=79.21, so the root is between 8.8 and 8.9.

**Final answer:** √78 ≈ 8.83.

### Method 3: Long Division

Pair the digits from the right and add decimal pairs: 78.0000.

Largest square ≤78 is 64=8²; first digit 8, remainder 14.

Bring down 000000 to make 1400; double the 8 to get 16, and find d so that 16d×d≤1400: 168×8=1344, so the next digit is 8, remainder 56.

Continue: the next digits give 8.832….

**Final answer:** √78 ≈ 8.832.

## Common Mistakes With Square Root of 78

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

**Where it slips in:** assuming every radical reduces to something smaller. **Don't do this:** write 78=3√26 or 2392 as if a factor came out. **The correct way:** check the prime factors first.

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

**Where it slips in:** writing 78=8.8 and treating it as exact. **Don't do this:** report 8.8 in a calculation that needs precision. **The correct way:** keep the exact form through the algebra and round only at the final step.

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

**Where it slips in:** expecting a clean whole-number answer. **Don't do this:** guess 78=8. **The correct way:** understand its root is irrational, sitting between 8 and 9.

## Where to Go From Here

Estimate √75 and √80 using the same methods above.

### Frequently Asked Questions

1. **Is the square root of 78 rational or irrational?**  
   Irrational. Prime factors have no repeated pairs.
2. **Can the square root of 78 be simplified?**  
   No. Already in simplest radical form.
3. **What is the square root of 78 to two decimal places?**  
   Approximately 8.83.
4. **What is the square root of 81?**  
   9, exactly.
5. **What is the square root of -78?**  
   There is no real square root; it involves imaginary numbers.