Cube root of 512 -Different methods

What do you mean by Cube root of 512?

When a number is multiplied by itself three times to give a product of 512, then that number is the cube root of 512.

512 = 8 × 8 × 8

Now we can see that the cube root of 512 is 8.

It can be also written as \sqrt[3]{512} \text { or } 512^{1 / 3}.

We are going to find out the cube root using two methods:

1. Prime factorisation method
2. Long division method

Prime factorization method

This is the easiest method to find out the cube roots but it becomes quite lengthy when we have to find cube roots of large numbers.

We have to follow the following steps

1. First, we have to do the prime factorization of 512. It is usually done by dividing the number by the smallest possible prime factor. In this case, we have to divide it by 2 first. We have to divide it by prime numbers until the last quotient is 1. It is done as follows.

2. Now we can write,

512=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2

3. After getting the prime factors we have to divide them into groups of threes containing the same factors.

Here the first group contains 2 × 2 × 2, the second group contains 2 × 2 × 2 and the third group contains 2 × 2 × 2.

512=(\underline{2 \times 2 \times 2}) \times(\underline{2 \times 2 \times 2}) \times(\underline{2 \times 2 \times 2})

4. Now we have to take 1 number from each group

Here we take 2 from the first group, 2 from the second group and 2 from the third group.

5. We have to multiply the factors we got from each group.
This product is the cube root of 512.

Estimation method

This method takes less time as compared to the previous one. We can use it for finding cube roots of large numbers that are perfect cubes.

We have to follow the following steps:

1. First, the number is divided by making groups of three digits from the right side.

If the last group on the left side doesn’t have 3 digits, it is okay and we will consider that group as it is.

One group is formed in this case as shown below. 512 is the first group

2. The unit place of the first group from the right determines the unit place of the cube root of the number.

Refer to the following lookup table 

So, the unit place of the first group 512 is 2. By referring to the above table, we know that the unit place of \sqrt[3]{512} is 8.

3. Therefore we can conclude that 8 is the value of \sqrt[3]{512}.

Solved Examples

1. What is the cube root of 4096?

Solution

4096 = 512 × 8

∛4096 = ∛(512 × 8 )

∛4096 = ∛(512  ) × ∛( 8 )

We already know that 8 is the cube root of 512 and 2 is the cube root of 8.

∛4096 = 8 × 2

∛4096 = 16

2. What is the length of each side of a cube if its volume is 512 m^{3} \text { ? }?

Solution

The volume of a cube = side × side × side

side = cube root of the volume 

        = \sqrt[3]{512}

        = 8 m