Cube Root of 100 — Value and Steps

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.

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 $n$ is the value that, multiplied by itself three times, gives $n$. In symbols, $\sqrt[3]{n} = x$ means $x^3 = n$. 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 $4^3 = 64$ and $5^3 = 125$, 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.

$100 = 2 \times 2 \times 5 \times 5$

$100 = 2^2 \times 5^2$

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 $\sqrt[3]{100}$.

Method 2: Estimation by bracketing

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

$4^3 = 64$

$5^3 = 125$

So $4 < \sqrt[3]{100} < 5$. Now test a value in between.

$4.6^3 = 97.336$

$4.7^3 = 103.823$

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

$4.64^3 = 99.897$

$4.65^3 = 100.545$

Final answer: $\sqrt[3]{100} \approx 4.642$.

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 $\sqrt[3]{100} = 10$ because $10^2 = 100$.

The correct way: the index is 3, so you need $x^3 = 100$, not $x^2 = 100$. 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 $\sqrt{100}$ when you mean the cube root.

The correct way: keep the index visible: $\sqrt[3]{100}$. A missing index silently changes the problem into a square root.

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 $\sqrt[3]{100} = 2\sqrt[3]{25}$ by pulling a 2 out.

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

Conclusion

• The cube root of 100 is irrational, with value $\sqrt[3]{100} \approx 4.642$.

• 100 factors as $2^2 \times 5^2$, 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.

To go further with radicals and roots alongside a teacher, explore Bhanzu's algebra tutor or browse math classes online.

Read More

• Cube root of 64 — a clean perfect-cube example to contrast with ∛100

• Cube root of 27 — another exact cube root

• Cube root of 343 — the cube root of 7³

• Cube numbers — the perfect cubes that make bracketing work

• Square root of 100 — the square-root counterpart, which is exactly 10

• What is a square root — the sibling operation to cube roots