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 3^{3} = 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 1^{3} = 1, 3^{3} = 27, 5^{3} = 125, ………….. are all odd.

**(ii)** True

Since a perfect cube ends with three zeroes.

Ex: 10^{3} = 1000, 20^{3} = 8000, 30^{3} = 27000, ………….. so on.

**(iii)** False.

Since 5^{2} = 25, 5^{3} = 125, 15^{2} = 225, 15^{3} = 3375 (Did not end with 25)

**(iv)** False

Since 12^{3} = 1728 (End with 8)

And 22^{3} = 10648 (End with 8)

**(v)** False

Since 10^{3} = 1000 (Four digit number)

And 11^{3} = 1331 (Four digit number)

**(vi)** False

Since 99^{3} = 970299 (Six digit number)

**(vii)** True

1^{3} = 1 (Single digit number)

2^{3} = 8 (Single digit number)

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