The cube root of 5 is $\sqrt[3]{5} \approx 1.710$, and because 5 is prime, the radical does not simplify.
Quick Answer:
Result: $\sqrt[3]{5} \approx 1.710$
Notation: $\sqrt[3]{5}$ or $5^{1/3}$
Method shown: estimation between neighbouring cubes
Approximate value: $1.710$ (to 3 decimal places, irrational)
Exact form: $\sqrt[3]{5}$ (already in simplest radical form)
Quick Reference Table
Number | Simplified cube root | Decimal (3 dp) |
|---|---|---|
$\sqrt[3]{1}$ | $1$ | $1.000$ |
$\sqrt[3]{3}$ | $\sqrt[3]{3}$ | $1.442$ |
$\sqrt[3]{4}$ | $\sqrt[3]{4}$ | $1.587$ |
$\sqrt[3]{5}$ | $\sqrt[3]{5}$ | $1.710$ |
$\sqrt[3]{6}$ | $\sqrt[3]{6}$ | $1.817$ |
$\sqrt[3]{8}$ | $2$ | $2.000$ |
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 $\sqrt[3]{5} \approx 1.710$, 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 $1^3 = 1$ and $2^3 = 8$.
The result is an irrational number that never terminates and never repeats. Because 5 is prime, it carries no perfect-cube factor, so $\sqrt[3]{5}$ is already in its simplest form, the same principle at work across cube numbers and radicals.
How to Find the Cube Root of 5 (Methods)
Method 1: Estimation between neighbouring cubes
Locate 5 between the two nearest perfect cubes.
$$1^3 = 1$$ $$2^3 = 8$$
So $\sqrt[3]{5}$ lies between 1 and 2. Since 5 is roughly midway, start near the middle. Testing $1.7$:
$$1.7^3 = 4.913$$
That is below 5, so try $1.72$:
$$1.72^3 \approx 5.088$$
That is just above 5, so the answer sits between 1.70 and 1.72. Testing $1.710$:
$$1.710^3 \approx 5.000$$
Final answer: $\sqrt[3]{5} \approx 1.710$ to three decimal places.
Method 2: Why it will not simplify
Write 5 in terms of its prime factors.
$$5 = 5$$
For a factor to leave a cube root, it must appear three times. The single 5 has no group of three.
$$\sqrt[3]{5} = \sqrt[3]{5}$$
Final answer: $\sqrt[3]{5}$ 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.
Don't do this: writing $\sqrt{5}$ when you mean the cube root.
The correct way: always show the index: $\sqrt[3]{5}$. Without the little 3 it reads as a square root, giving $\approx 2.236$ instead of $\approx 1.710$.
Mistake 2: Trying to simplify a prime radical
Where it slips in: assuming every cube root breaks down like $\sqrt[3]{40} = 2\sqrt[3]{5}$.
Don't do this: writing $\sqrt[3]{5}$ as a product of smaller radicals.
The correct way: a prime number under a cube root has no perfect-cube factor, so it stays as $\sqrt[3]{5}$.
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.
Don't do this: answering roughly 1.9 or 2.
The correct way: cube roots grow slowly. Since $1.7^3 = 4.913$, the answer is close to 1.71, not 2.
To build cube-root fluency with a teacher, explore Bhanzu's algebra tutor or math classes online.
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