Cube Root of 24 — Value and Simplification Steps
Cube Root of 24 — Value and Simplification Steps
TL;DR
The cube root of 24 simplifies to (2\sqrt[3]{3}), with a decimal value of about 2.884. This article shows the prime-factorization simplification, the estimation method, where it appears, common mistakes, and worked examples.
Quick Reference Table
| Number | Simplified cube root | Decimal (3 dp) |
|---|---|---|
| (8\sqrt[3]{8}) | (2^2) | 2.000 |
| (16\sqrt[3]{16}) | (2^2\sqrt[3]{2}) | 2.520 |
| (24\sqrt[3]{24}) | (2^3\sqrt[3]{3}) | 2.884 |
| (27\sqrt[3]{27}) | (3) | 3.000 |
| (54\sqrt[3]{54}) | (3^2\sqrt[3]{2}) | 3.780 |
| (64\sqrt[3]{64}) | (4) | 4.000 |
Where the Cube Root of 24 Appears
The cube root of 24 turns up whenever a volume is known and an edge length is wanted. If a cube-shaped tank holds 24 cubic units, each edge measures about (2.884) units. The same simplification pattern — pulling a perfect-cube factor out from under the radical — is what scientific and engineering calculators do internally before rounding.
What Is the Cube Root of 24?
The cube root of a number is the value that, multiplied by itself three times, returns that number. Since 24 is not a perfect cube — it sits between (8) and (27) — its cube root is an irrational number that never terminates.
But it is not fully "stuck" under the radical. Because 24 contains the perfect-cube factor 8, part of it can come out, leaving the simplified form (2\sqrt[3]{3}). This is the same skill used across cube numbers and radicals.
How to Simplify the Cube Root of 24 (Methods)
Method 1: Prime factorization
Break 24 into its prime factors.
[24=2 imes 2 imes 2 imes 3 = 2^3 imes 3]
Write the cube root over the factored form.
[\sqrt[3]{24} = \sqrt[3]{2^3 \times 3}]
The cube root of (2^3) is 2, so 2 comes out of the radical while the 3 stays inside.
[\sqrt[3]{24} = 2\sqrt[3]{3}]
Final answer: (\sqrt[3]{24} = 2\sqrt[3]{3}).
Method 2: Estimation for the decimal value
Locate 24 between neighbouring perfect cubes.
[\sqrt[3]{24} \text{ lies between } 2 \text{ and } 3, \text{ and close to } 3 \text{ because } 24 \text{ is close to } 27.]
Testing 2.9:
[2.9^3 = 24.389]
That is slightly above 24, so the answer is a touch below 2.9. Testing 2.88:
[2.88^3 \approx 23.888]
Final answer: (\sqrt[3]{24} \approx 2.884) to three decimal places.
Common Mistakes With Cube Root of 24
Mistake 1: Dropping the cube-root index
Where it slips in: writing the radical quickly.
Don't do this: writing (\sqrt{24}) when you mean the cube root.
The correct way: always show the index: (\sqrt[3]{24}). Without the little 3 it reads as a square root, giving approximately 4.899 instead of approximately 2.884.
Mistake 2: Pulling out the wrong factor
Where it slips in: simplifying by removing any factor instead of a perfect-cube factor.
Don't do this: writing (4\sqrt[3]{6}) by splitting off 4.
The correct way: only a perfect cube factor can leave the radical. Here that is 8, giving (2\sqrt[3]{3}).
Mistake 3: Treating the cube root like a square root when simplifying
Where it slips in: grouping factors in pairs instead of triples.
Don't do this: taking one factor out for every pair of 2's.
The correct way: a cube root removes a factor for every three copies. Three 2's give one 2 outside; the leftover 3 has no triple, so it stays inside.
Frequently Asked Questions
What is the cube root of 24 in simplest radical form?
It is (2\sqrt[3]{3}), found by pulling the perfect-cube factor (8) out of 24.
Is the cube root of 24 rational or irrational?
Irrational. Since 24 is not a perfect cube, (\sqrt[3]{24}) is a non-terminating, non-repeating decimal, about 2.884.
What is the cube root of 24 as a decimal?
Approximately 2.884 to three decimal places.
Why does 2 come out of (\sqrt[3]{24}) but 3 does not?
Because (24=2^3 \times 3). The factor (2^3) is a perfect cube, so its cube root is the whole number 2; the lone 3 has no cube factor, so it remains under the radical.
How is the cube root of 24 different from the cube root of 27?
(\sqrt[3]{27} = 3) exactly, because 27 is a perfect cube. (\sqrt[3]{24}) is irrational and simplifies only to (2\sqrt[3]{3}.