Class 8 Maths Cubes and Cube Roots Cube Roots

Cube Roots

We know that finding the square root is the inverse operation of squaring. Similarly, finding the cube root is the inverse operation of finding cube.

We know that 33 = 27; so we say that the cube root of 27 is 3.

We write 3√27 = 3.

The cube-root is denoted by the symbol 3√.

Here is a list of some numbers and its cube roots. Cube root through prime factorization method

We find its cube root by prime factorization method. Let us take some examples to find the cube roots using prime factorization method.

Problem: Find the cube root of each of the following numbers by prime factorization method:

(i) 64    (ii) 512   (iii) 10648

Solution:

(i) 64

3√64 = √(2 * 2 * 2 * 2 * 2 * 2)

3√64 = 2 * 2

3√64 = 4 (ii) 512

3√512 = √(2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2)

3√512 = 2 * 2 * 2

3√512 = 8 (iii) 10648

3√10648 = √(2 * 2 * 2 * 11 * 11 * 11)

3√10648 = 2 * 11

3√10648 = 22 Cube root of a cube number

Till now, we have seen cube of many numbers. If we know that the given number is a cube number then We can find its cube root through estimation method.

Let us take some examples.

Example 1: Let a cube number is 13824

To calculate the cube root of the number 133824, we will do the following steps:

Step 1. Form groups of three starting from the rightmost digit of 13824 i.e. groups are 13 and 824.

In this case, 824 has three digits whereas 13 has only two digits.

Step 2. The First group, i.e. 824 will give us the unit’s digit of the required cube root.

The number ends with digit 4. We know that 4 comes at the unit’s place of a number only when it’s cube root ends in 4.

So, we get 4 at the unit’s place of the cube root.

Step 3. Take another group i.e. 13

Now, cube of 2 is 8 and cube of 3 is 27. Then number 13 lies between 8 and 27.

The smaller number among 2 and 3 is 2.

The one’s place of 2 is 2 itself. Take 2 as ten’s place of the cube root of 17576.

Thus 3√13824 = 24

Example 2. Let us take another cube number 110592.

To calculate the cube root of the number 110592, we will do the following steps:

Step 1. Form groups of three starting from the rightmost digit of 110592 i.e. groups are 110 and 592.

Step 2. The First group, i.e. 592 will give us the unit’s digit of the required cube root.

The number ends with digit 2. We know that 2 comes at the unit’s place of a number only when it’s cube  root ends in 8.

So, we get 8 at the unit’s place of the cube root.

Step 3. Take another group i.e. 110

Now, cube of 4 is 64 and cube of 5 is 125. Then number 110 lies between 64 and 125.

The smaller number among 4 and 5 is 4.

Take 4 as ten’s place of the cube root of 110592.

Thus 3√110592 = 48

So, in this way, we can find the cube root of a number using estimation method.

Problem: State true or false:

(i) Cube of any odd number is even.

(ii) A perfect cube does not end with two zeroes.

(iii) If square of a number ends with 5, then its cube ends with 25.

(iv) There is no perfect cube which ends with 8.

(v) The cube of a two digit number may be a three digit number.

(vi) The cube of a two digit number may have seven or more digits.

(vii) The cube of a single digit number may be a single digit number.

Solution:

(i) False

Since 13 = 1, 33 = 27, 53 = 125, ………….. are all odd.

(ii) True

Since a perfect cube ends with three zeroes.

Ex: 103 = 1000, 203 = 8000, 303 = 27000, ………….. so on.

(iii) False.

Since 52 = 25, 53 = 125, 152 = 225, 153 = 3375 (Did not end with 25)

(iv) False

Since 123 = 1728 (End with 8)

And 223 = 10648 (End with 8)

(v) False

Since 103 = 1000 (Four digit number)

And 113 = 1331 (Four digit number)

(vi) False

Since 993 = 970299 (Six digit number)

(vii) True

13 = 1 (Single digit number)

23 = 8 (Single digit number)

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