Cube Root Calculator

Information about Cube Roots

The cube root of a number is a value that, when multiplied by itself twice (or cubed), gives the original number. It is denoted by the symbol \sqrt[3]{}.

For example, the cube root of 8 is 2 because 2 times 2 times 2 equals 8. Unlike square roots, cube roots only have one real solution (either positive or negative).

\sqrt[3]{8} = 2

How to Find the Cube Root of a Number Without a Calculator?

1. Start by guessing two numbers whose cube roots you know, ensuring that your number is between them.
2. Divide your number by the square of one of the guesses you made in step 1.
3. Take the number you got from step 2 and your initial guess, then find the midpoint between them.
4. Use the midpoint to go back to step 2 and repeat the process until you get a number close enough to the desired accuracy.

Let's Find the Cube Root of 20

Let's figure out the cube root of 20, aiming to get it accurate to two decimal places.

Step 1: We need to find two numbers whose cube roots we know, and 20 should be between them. We know that:

2 × 2 × 2 = 8

3 × 3 × 3 = 27

So, the cube root of 20 is between 2 and 3.

Step 2: Let's divide 20 by the square of the closer number, which is 2:

20 ÷ (2 × 2) = 20 ÷ 4 = 5

Step 3: Now, let's find the midpoint between 5 and 2. Add them together and divide by 2:

(5 + 2) ÷ 2 = 3.5

So, our new guess is 3.5.

Step 4: Let's repeat the process with 3.5:

First, divide 20 by the square of 3.5:

20 ÷ (3.5 × 3.5) = 20 ÷ 12.25 ≈ 1.63

Now, find the midpoint between 1.63 and 3.5:

(1.63 + 3.5) ÷ 2 ≈ 2.565

So now, we guess 2.565.

Step 5: Let's see if 2.565 is close enough:

Multiply 2.565 by itself twice:

2.565 × 2.565 × 2.565 ≈ 16.88

This is fairly close to 20, so the cube root of 20 is approximately 2.565.

If you're happy with this answer, you can stop here! If you want to be even more precise, you could repeat the steps again.

Some common list of perfect cube roots

∛1 = 1, since 1 × 1 × 1 = 1

∛8 = 2, since 2 × 2 × 2 = 8

∛27 = 3, since 3 × 3 × 3 = 27

∛64 = 4, since 4 × 4 × 4 = 64

∛125 = 5, since 5 × 5 × 5 = 125

∛216 = 6, since 6 × 6 × 6 = 216

∛343 = 7, since 7 × 7 × 7 = 343

∛512 = 8, since 8 × 8 × 8 = 512

∛729 = 9, since 9 × 9 × 9 = 729

∛1000 = 10, since 10 × 10 × 10 = 1000

∛1331 = 11, since 11 × 11 × 11 = 1331

∛1728 = 12, since 12 × 12 × 12 = 1728

∛2197 = 13, since 13 × 13 × 13 = 2197

∛2744 = 14, since 14 × 14 × 14 = 2744

∛3375 = 15, since 15 × 15 × 15 = 3375

∛4096 = 16, since 16 × 16 × 16 = 4096

∛4913 = 17, since 17 × 17 × 17 = 4913

∛5832 = 18, since 18 × 18 × 18 = 5832

∛6859 = 19, since 19 × 19 × 19 = 6859

∛8000 = 20, since 20 × 20 × 20 = 8000

List of imperfect cube roots

∛2 ≈ 1.25992104989

∛3 ≈ 1.44224957031

∛5 ≈ 1.70997594668

∛6 ≈ 1.81712059283

∛7 ≈ 1.91293118277

∛10 ≈ 2.15443469003

∛11 ≈ 2.22398009057

∛12 ≈ 2.28942848511

∛13 ≈ 2.35133468772

∛14 ≈ 2.41014226418

∛15 ≈ 2.46621207433

∛17 ≈ 2.57128159066

∛18 ≈ 2.62074139421

∛19 ≈ 2.66840164872

∛20 ≈ 2.71441761659

∛21 ≈ 2.75892417638

∛22 ≈ 2.80203933066

∛23 ≈ 2.84386697985

∛24 ≈ 2.88449914061

∛26 ≈ 2.96249606841

∛50 ≈ 3.68403149864

∛80 ≈ 4.30886938006

∛90 ≈ 4.48140474604

∛99 ≈ 4.62606500918

∛101 ≈ 4.63480768213