# Cube Root of 100 — Value and Steps

TL;DR

The cube root of 100 is an irrational number equal to ∛100 ≈ 4.642, because 100 factors as 2² × 5² with no perfect cube inside it. This article gives the value, the reason ∛100 cannot be simplified, two ways to compute it by hand, and the mistakes to avoid.

**The cube root of 100 is approximately** **4.642**. Written exactly it stays as **∛100**, since 100 has no perfect-cube factor to pull out, so the decimal never terminates and never repeats.

> **Quick Answer:**  
> **Result:** ∛100 ≈ 4.642  
> **Notation:** Radical form ∛100; decimal form 4.6416 (to 4 dp)  
> **Method shown:** Prime factorisation to test for cube factors, then estimation by bracketing  
> **Approximate value:** 4.6416 (irrational, non-terminating)  
> **Exact form:** ∛100 (cannot be simplified to a whole number or a smaller radical)

## Quick Reference Table of Nearby Cube Roots

The table below sits ∛100 among its neighbours so you can see the spacing between cube roots and check the estimate.

| Number nnn | Cube root n3\sqrt[3]{n}3n​ | Type |  
| --- | --- | --- |  
| 64 | 643=4\sqrt[3]{64} = 4364​=4 | Exact (perfect cube) |  
| 100 | 1003≈4.642\sqrt[3]{100} \approx 4.6423100​≈4.642 | Irrational |  
| 125 | 1253=5\sqrt[3]{125} = 53125​=5 | Exact (perfect cube) |  
| 200 | 2003≈5.848\sqrt[3]{200} \approx 5.8483200​≈5.848 | Irrational |  
| 216 | 2163=6\sqrt[3]{216} = 63216​=6 | Exact (perfect cube) |  
| 1000 | 10003=10\sqrt[3]{1000} = 1031000​=10 | Exact (perfect cube) |

The two perfect cubes on either side of 100 are **64** and **125**, so ∛100 must land between 4 and 5, closer to 5.

## What a Cube Root Means

The cube root of a number nnn is the value that, multiplied by itself three times, gives nnn. In symbols, n3=x\sqrt[3]{n} = x3n​=x means x3=nx^3 = nx3=n. The small **3** tucked into the radical sign is the _index_; it is what separates a cube root from a square root, and it must always be written for a cube root.

Because 43=644^3 = 6443=64 and 53=1255^3 = 12553=125, and 100 sits between them, ∛100 is a number between 4 and 5 that is not a whole number. That makes it **irrational** — its decimal expansion runs forever without repeating.

## How to Compute the Cube Root of 100

### **Method 1: Prime factorisation (the simplify test)**

Break 100 into primes to check for a perfect-cube factor.

100=2×2×5×5100 = 2 \times 2 \times 5 \times 5100=2×2×5×5

100=22×52100 = 2^2 \times 5^2100=22×52

A cube root simplifies only when a prime appears **three** times (or in a multiple of three). Here the 2 appears twice and the 5 appears twice — no prime reaches a group of three.

**Final answer:** ∛100 has no perfect-cube factor, so it cannot be simplified. It stays as 1003\sqrt[3]{100}3100​.

### **Method 2: Estimation by bracketing**

Trap the value between two cubes you know, then narrow it.

43=644^3 = 6443=64  
53=1255^3 = 12553=125

So 4<1003<54 < \sqrt[3]{100} < 54<3100​<5. Now test a value in between.

4.63=97.3364.6^3 = 97.3364.63=97.336

4.73=103.8234.7^3 = 103.8234.73=103.823

Since 100 sits between 97.336 and 103.823, the answer is between 4.6 and 4.7, and nearer 4.6.

4.643=99.8974.64^3 = 99.8974.643=99.897

4.653=100.5454.65^3 = 100.5454.653=100.545

**Final answer:** 1003≈4.642\sqrt[3]{100} \approx 4.6423100​≈4.642.

## Common Mistakes With Cube Root of 100

### **Mistake 1: Confusing the cube root with the square root**

**Where it slips in:** reading ∛100 as √100 and answering 10.

**Don't do this:** write 1003=10\sqrt[3]{100} = 103100​=10 because 102=10010^2 = 100102=100.

**The correct way:** the index is 3, so you need x3=100x^3 = 100x3=100, not x2=100x^2 = 100x2=100. The answer is about 4.642, not 10. Students first meeting radicals often ignore the little index and default to square roots — always read the index first.

### **Mistake 2: Dropping the index when writing the answer**

**Where it slips in:** copying the radical as a plain √ during a longer calculation.

**Don't do this:** write 100\sqrt{100}100​ when you mean the cube root.

**The correct way:** keep the index visible: 1003\sqrt[3]{100}3100​.

### **Mistake 3: Trying to force a simplification that isn't there**

**Where it slips in:** assuming every radical breaks into a smaller radical.

**Don't do this:** claim 1003=2253\sqrt[3]{100} = 2\sqrt[3]{25}3100​=2325​ by pulling a 2 out.

**The correct way:** you can only pull out a factor that is itself a perfect cube. Since 100=22×52100 = 2^2 \times 5^2100=22×52 has no cube factor, nothing comes out. 1003\sqrt[3]{100}3100​ is already in simplest form.

## Conclusion

- The **cube root of 100** is irrational, with value 1003≈4.642\sqrt[3]{100} \approx 4.6423100​≈4.642.

- 100 factors as 22×522^2 \times 5^222×52, so it has no perfect-cube factor and ∛100 cannot be simplified.

- The value sits between 4 and 5 because 100 lies between the cubes 64 and 125.

- Bracketing between known cubes gives a reliable hand estimate without a calculator.

## Frequently Asked Questions

**Is the cube root of 100 rational or irrational?**  
Irrational. 100 is not a perfect cube, so ∛100 cannot be written as a fraction and its decimal never ends.

**What is the cube root of 100 to two decimal places?**  
About 4.64.

**Can ∛100 be simplified into a smaller radical?**  
No. Its prime factorisation is 22×522^2 \times 5^222×52, and no prime appears three times, so there is no perfect-cube factor to remove.

**What is the difference between ∛100 and √100?**  
100=10\sqrt{100} = 10100​=10 because 102=10010^2 = 100102=100, while 1003≈4.642\sqrt[3]{100} \approx 4.6423100​≈4.642 because that value cubed gives 100. The index changes the answer completely.

**What two whole numbers is the cube root of 100 between?**  
Between 4 and 5, since 43=644^3 = 6443=64 and 53=1255^3 = 12553=125.