Cube Root of 100 — Value and Steps
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.