# Square Root of 64 - Value, Calculation, Examples

## TL;DR

The square root of 64 (\(\sqrt{64}\)) equals exactly 8 — no decimal expansion, no irrationality, just a clean whole number. This article covers why \(64=8\), three methods to compute it, where 64 shows up as a perfect square in computing and geometry, and the most common mistakes.

## The square root of 64

The **square root of 64** is exactly 8, because \(8 \times 8 = 64\). Unlike \(\sqrt{10}\) or \(\sqrt{12}\), \(64\) is a _perfect square_.

### Quick Answer:

- **Result:** \(64=8\)
- **Notation:** \(\sqrt{64} = 8\) (exact integer)
- **Method shown:** Prime factorization
- **Exact form:** 8 — 64 is a perfect square (\(8^2 = 64\))
- **Irrational?** No — rational, in fact a whole number

## Quick Reference Table

| Number n | \(\sqrt{n}\) | 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 |
| 121 | 11  | Yes |
| 144 | 12  | Yes |

## Where 64 Appears

The number 64 itself shows up everywhere in **binary computing** — \(64=2^6\), so a 64-bit processor uses 64 binary digits per word, and "64-bit color" means \(2^{64}\) possible color values. The square root of 64 — that's 8 — is the side length of an **8×8 chessboard**, which has 64 squares total. And in music, 64 appears as the number of subdivisions in some Indian classical rhythmic cycles ( _talas_).

## What Is a Perfect Square?

A **perfect square** is a non-negative integer that is the square of another integer. The first few perfect squares are:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100,…

These come from 1², 2², 3², 4²,… — squaring each natural number.

64 is the 8th perfect square (\(8^2 = 64\)). Its square root — by definition — is the integer 8. No decimal, no approximation.

## Three Methods to Find \(\sqrt{64}\)

### Method 1: Prime Factorization

#### Step 1: Find the prime factorization of 64.

\[64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6\]

#### Step 2: Pair the prime factors.

\[64 = (2 \times 2) \times (2 \times 2) \times (2 \times 2) = (2^2) \times (2^2) \times (2^2)\]

#### Step 3: Take one factor from each pair under the square root.

\[\sqrt{64} = 2 \times 2 \times 2 = 8\]

This works for any perfect square — if you can pair every prime factor, the result is an integer.

### Method 2: Direct Recall

Memorise the first 12–15 perfect squares. \(64=8\) is one of the most-used.

### Method 3: Repeated Subtraction

64 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15.

Count the odd numbers: 1, 3, 5, 7, 9, 11, 13, 15 — that's 8 numbers. So \(\sqrt{64} = 8\).

### Method 4: Long Division

The long-division algorithm works for any number, not only perfect squares.

#### Step 1: Pair digits of 64 from the decimal point.

#### Step 2: Find the greatest integer whose square is \(\leq 64\). That's 8.

#### Step 3: Subtract: 64 − 64 = 0. No remainder — the algorithm halts.

**Quotient:** \(\sqrt{64} = 8\) exactly.

## Is the Square Root of 64 Rational or Irrational?

\(\sqrt{64} = 8\), an integer. **Rational** — every integer is rational.

## Common Mistakes with Square Root of 64

**Mistake 1:** Writing \(64 = \pm 8\) in the wrong context. The principal square root \(\sqrt{64}\) is 8 (positive only).

**Mistake 2:** Confusing \(\sqrt{64}\) with the cube root.

**Mistake 3:** Treating \(\sqrt{64}\) as needing approximation. Recognise perfect squares on sight and skip the calculator.

## Frequently Asked Questions

**What is the value of \(\sqrt{64}\)?** 8 exactly.

**Is 64 a perfect square?** Yes, it is the 8th perfect square.

**Is the square root of 64 rational or irrational?** Rational, it is an integer.

**What is the cube root of 64?** \(4\), because \(4^3 = 64\).

**How do I find the square root of 64 without a calculator?** Recognise it as a perfect square, or use prime factorization.