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

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.