Cube Root of 5 — Value and Steps
Cube Root of 5 — Value and Steps
TL;DR
The cube root of 5 is 53≈1.710, an irrational number that cannot be simplified because 5 is prime. This article shows why the radical stays as is, how to estimate its value by hand, where it appears, common mistakes, and worked examples.
The cube root of 5 is 53≈1.710, and because 5 is prime, the radical does not simplify.
Quick Answer:
Result: 53≈1.710
Notation: 53 or 51/3
Method shown: estimation between neighbouring cubes
Approximate value: 1.710 (to 3 decimal places, irrational)
Exact form: 53 (already in simplest radical form)
Quick Reference Table
| Number | Simplified cube root | Decimal (3 dp) |
|---|---|---|
| 13 | 1 | 1.000 |
| 33 | 33 | 1.442 |
| 43 | 43 | 1.587 |
| 53 | 53 | 1.710 |
| 63 | 63 | 1.817 |
| 83 | 222 | 2.000 |
Where the Cube Root of 5 Appears
The cube root of 5 is the edge length of a cube whose volume is 5 cubic units. It also appears in scaling problems: a cube-shaped container whose capacity must grow five times, while staying a cube, needs each edge multiplied by 53≈1.710, not by 5.
What Is the Cube Root of 5?
The cube root of a number is the value that, multiplied by itself three times, returns that number. Since no whole number cubed equals 5, the cube root of 5 sits between 1 and 2.
The result is an irrational number that never terminates and never repeats. Because 5 is prime, it carries no perfect-cube factor, so 53 is already in its simplest form.
How to Find the Cube Root of 5 (Methods)
Method 1: Estimation between neighbouring cubes
Locate 5 between the two nearest perfect cubes.
So 53 lies between 1 and 2. Since 5 is roughly midway, start near the middle. Testing 1.7:
That is below 5, so try 1.72:
That is just above 5, so the answer sits between 1.70 and 1.72. Testing 1.710:
Final answer: 53≈1.710 to three decimal places.
Method 2: Why it will not simplify
Write 5 in terms of its prime factors.
For a factor to leave a cube root, it must appear three times. The single 5 has no group of three.
Final answer: 53 is already in simplest radical form.
Common Mistakes With Cube Root of 5
Mistake 1: Dropping the cube-root index
Where it slips in: writing the radical quickly.
The correct way: always show the index: 53. Without the little 3 it reads as a square root, giving ≈2.236 instead of ≈1.710.
Mistake 2: Trying to simplify a prime radical
Where it slips in: assuming every cube root breaks down like 40.
The correct way: a prime number under a cube root has no perfect-cube factor, so it stays as 53.
Mistake 3: Guessing the value is close to 2
Where it slips in: noticing 5 is near 8 and assuming the cube root is near 2.
The correct way: cube roots grow slowly, the answer is close to 1.71, not 2.
Frequently Asked Questions
What is the cube root of 5?
The cube root of 5 is 53≈1.710, an irrational number.
Can the cube root of 5 be simplified?
No. Because 5 is prime, it has no perfect-cube factor, so 53 is already in simplest radical form.
Is the cube root of 5 rational or irrational?
Irrational. Since 5 is not a perfect cube, 53 is a non-terminating, non-repeating decimal.
What is the cube root of 5 in exponential form?
It is 51/3, which is the same as 53.