General (Nth) Root Calculator

Information about Nth Roots

The nth root of a number is a value that, when multiplied by itself n times, gives the original number. It is denoted by the symbol \sqrt[n]{}.

For example, the cube root of 8 is 2 because 2 times 2 times 2 equals 8. The nth root can be positive or negative, depending on the number and the root.

\sqrt[3]{8} = 2

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

1. Start by guessing two numbers whose nth roots you know, ensuring that your number is between them.
2. Divide your number by the (n-1)th power 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 5th Root of 32

Let's figure out the 5th root of 32, aiming to get it accurate to two decimal places.

Step 1: We need to find two numbers whose 5th roots we know, and 32 should be between them. We know that:

2⁵ = 32

So, the 5th root of 32 is exactly 2.

Step 2: If we didn't know it was exactly 2, let's divide 32 by the 4th power of the closer number, which is 2:

32 ÷ (2 × 2 × 2 × 2) = 32 ÷ 16 = 2

Step 3: Now, let's find the midpoint between 2 and 2. Since they're the same, the midpoint is also 2.

So, our guess remains 2.

Step 4: Repeat the process if needed. However, since 32 is a perfect 5th power of 2, the process stops here.

If the number was not a perfect power, we would continue refining our guess until satisfied.

Let's Find the 4th Root of 100

Step 1: Find two numbers whose 4th roots you know, and 100 should be between them. We know that:

2⁴ = 16

3⁴ = 81

4⁴ = 256

So, the 4th root of 100 is between 3 and 4.

Step 2: Divide 100 by the 3rd power of the closer number, which is 3:

100 ÷ (3 × 3 × 3) = 100 ÷ 27 ≈ 3.70

Step 3: Now, let's find the midpoint between 3.70 and 3:

(3.70 + 3) ÷ 2 = 3.35

So, our new guess is 3.35.

Step 4: Repeat the process with 3.35 to get closer:

Divide 100 by 3.35³:

100 ÷ 37.63 ≈ 2.66

Average 2.66 and 3.35 to get a new guess:

(2.66 + 3.35) ÷ 2 ≈ 3.01

If this is close enough, you can stop. Otherwise, repeat the steps to get even more accurate.

Some list of perfect common nth roots

√1 = 1, since 1 × 1 = 1

√4 = 2, since 2 × 2 = 4

√9 = 3, since 3 × 3 = 9

√16 = 4, since 4 × 4 = 16

√25 = 5, since 5 × 5 = 25

∛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

∜16 = 2, since 2 × 2 × 2 × 2 = 16

∜81 = 3, since 3 × 3 × 3 × 3 = 81

∜256 = 4, since 4 × 4 × 4 × 4 = 256

∜625 = 5, since 5 × 5 × 5 × 5 = 625

5th√32 = 2, since 2 × 2 × 2 × 2 × 2 = 32

5th√243 = 3, since 3 × 3 × 3 × 3 × 3 = 243

5th√1024 = 4, since 4 × 4 × 4 × 4 × 4 = 1024

5th√3125 = 5, since 5 × 5 × 5 × 5 × 5 = 3125

List of imperfect common nth roots

√2 ≈ 1.414, since 1.414 × 1.414 ≈ 2

√3 ≈ 1.732, since 1.732 × 1.732 ≈ 3

√5 ≈ 2.236, since 2.236 × 2.236 ≈ 5

∛2 ≈ 1.260, since 1.260 × 1.260 × 1.260 ≈ 2

∛5 ≈ 1.710, since 1.710 × 1.710 × 1.710 ≈ 5

∛10 ≈ 2.154, since 2.154 × 2.154 × 2.154 ≈ 10

∜20 ≈ 2.114, since 2.114 × 2.114 × 2.114 × 2.114 ≈ 20

∜50 ≈ 2.659, since 2.659 × 2.659 × 2.659 × 2.659 ≈ 50

5th√10 ≈ 1.585, since 1.585 × 1.585 × 1.585 × 1.585 × 1.585 ≈ 10

5th√50 ≈ 2.114, since 2.114 × 2.114 × 2.114 × 2.114 × 2.114 ≈ 50

5th√100 ≈ 2.512, since 2.512 × 2.512 × 2.512 × 2.512 × 2.512 ≈ 100