# General (Nth) Root Calculator

Free radical equation calculator. Solve radical equations step by step easily. Just enter the radical and radicand into the calculator and get step by step answer.

## 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

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.

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

- Start by guessing two numbers whose nth roots you know, ensuring that your number is between them.
- Divide your number by the (n-1)th power of one of the guesses you made in step 1.
- Take the number you got from step 2 and your initial guess, then find the midpoint between them.
- 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:

2^{5} = 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:

2^{4} = 16

3^{4} = 81

4^{4} = 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 3.35^{3}:

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

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