# Cube Root of 64 - Value, Method, Examples

## TL;DR
The cube root of 64 is exactly 4, since 4^3 = 4 × 4 × 4 = 64. Because 64 is a perfect cube, its cube root is a whole number — no decimal, no approximation. The simplified radical form is \(\sqrt[3]{64} = 4\).

## Quick Reference Table

| Quantity | Value |
| --- | --- |
| \( \sqrt[3]{64} \) | 4 exactly |
| 64 as a cube | 4^3 |
| Prime factorisation | 64 = (2^2)^3 |
| Decimal value | 4.000 (exact) |
| Is 64 a perfect cube? | **Yes** |
| Is \( \sqrt[3]{64} \) rational? | **Yes** — it's the integer 4 |

## What Is a Cube Root?
The **cube root** of a number n is a value r such that:

\[ r^3 = n \]

In words: _"what number, multiplied by itself three times, equals n?"_

Written as \( \sqrt[3]{n} \) or \( n^{1/3} \). For n=64, we want r such that \( r^3 = 64 \).

Trying small integers: 2^3 = 8 (too small), 3^3 = 27 (too small), 4^3 = 64 ✓. So \( \sqrt[3]{64} = 4 \).

## Where \( \sqrt[3]{64} = 4 \) Appears
The number 64 shows up in nature, technology, and everyday measurement more than you might expect:
- **Chess** has 8×8 = 64 squares.
- **Genetics** — the genetic code has 64 possible codons.
- **Computing** — 64-bit architectures dominate modern processors.
- **Volume calculations** — a cube with side 4 cm has volume 64 cm³.
- **3D printing** and architecture related to cubic volumes.

## Why 64 Is a Perfect Cube
A **perfect cube** is a number that's the cube of an integer. The first few perfect cubes:

1, 8, 27, 64, 125, …

64 is the **4th perfect cube** (4^3). Its cube root is the integer 4 — no decimals, no approximation needed.

## Three Methods to Find \( \sqrt[3]{64} \)

### Method 1: Prime Factorisation
1. Find the prime factorisation of 64: \( 64 = 2^6 \).
2. Group the factors into triples: \( 64 = (2^3) × (2^3) \).
3. Take one factor from each triple: \( 4 = 2 × 2 \).

### Method 2: Direct Recall
Memorise the first few perfect cubes: 1, 8, 27, 64, … Then \( \sqrt[3]{64} = 4 \).

### Method 3: Estimation by Bracketing
Bracket it:
- 3^3 = 27 (less than 64)
- 4^3 = 64 (equal)
- 5^3 = 125 (greater)

## Is the Cube Root of 64 Rational or Irrational?
\( \sqrt[3]{64} = 4 \) — an integer, which is rational (the ratio 4/1).

## Three Worked Examples — Quick, Standard, Stretch
### Quick — Direct Identification
Find \( \sqrt[3]{64} \).

Since 4 × 4 × 4 = 64, \( \sqrt[3]{64} = 4 \).

### Standard — Cube Root in an Expression
Simplify \( \sqrt[3]{64 \cdot x^3} \).

So \( \sqrt[3]{64} \cdot \sqrt[3]{x^3} = 4x \).

### Stretch — Volume Problem
A cube has volume 64 cm³. What is the side length?

\[ s = \sqrt[3]{64} = 4 \text{ cm} \]

## Common Mistakes of Cube Root of 64
### Mistake 1: Confusing cube root with square root
**Where it slips in:** Writing \( 64 = 4 \sqrt{64} \).
**The fix:** Distinguish cube roots from square roots.

### Mistake 2: Writing \( \sqrt[3]{64} = \pm 4 \)
**The fix:** Understand cube roots have a unique real value.

### Mistake 3: Forgetting the cube root of variables
Ensure correct simplification of variables.

## Frequently Asked Questions
- **Q: What is the cube root of 64?** The cube root of 64 is exactly **4**.
- **Q: Is the cube root of 64 a whole number?** Yes, it's an integer.
- **Q: How do you simplify the cube root of 64?** It simplifies to the integer 4.
- **Q: What is the cube root of -64?** \( -\sqrt[3]{-64} = -4 \).
- **Q: How does the cube root of 64 differ from the square root of 64?** Cube root is 4, square root is 8.
- **Q: Why is the cube root of 64 useful?** Applied in volume problems and computing.

## Key Takeaways
- **\( \sqrt[3]{64} = 4 \)** — no decimal needed.
- **64 is the 4th perfect cube**.
- **Three methods** to find it: prime factorisation, direct recall, estimation.
- **Rational** — every integer is rational.

## A Practical Next Step
Try finding these:
1. Find \( \sqrt[3]{125} \).
2. Find \( \sqrt[3]{-27} \).
3. What is the edge length of a cube with volume 64 m³?