# Cube and Cube Roots

## Exercise 7.2

Question 1: Find the cube root of each of the following numbers by prime factorisation method.

(i) 64

**Answer:** 64 = 2 × 2 × 2 × 2 × 2 × 2

= 2^{3} × 2^{3}

So, cube root of 64 = 2 × 2 = 4

(ii) 512

**Answer:** `512 = 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2`

`= 2^3 xx 2^3 xx 2^3`

So, cube root of 512 `=2xx2xx2=8`

(iii) 10648

**Answer:** `10648 = 2 xx 2 xx 2 xx 11 xx 11 xx 11`

`= 2^3 xx 11^3`

So, cube root of 10648 `=2xx11=22`

(iv) 27000

**Answer:** `27000 = 2 xx 2 xx 2 xx 3 xx 3 xx 3 xx 5 xx 5 xx 5`

`= 2^3 xx 3^3 xx 5^3`

So, sube root of 27000 `=2xx3xx5=30`

(v) 15625

**Answer:** `15625 = 5 xx 5 xx 5 xx 5 xx 5 xx 5`

`= 5^3 xx 5^3`

So, cube root of 15625 `=5xx5=25`

(vi) 13824

**Answer:** `13824 = 2 xx 2 xx 2 xx 2 xx 2 xx 2`` xx 2 xx 2 xx 2 xx 3 xx 3 xx 3`

`= 2^3 xx 2^3 xx 2^3 xx 3^3`

So, cube root of 13824 `=2xx2xx2xx3=24`

(vii) 110592

**Answer:** `110592 = 2^3 xx 2^3 xx 2^3 xx 2^3 xx 3^3`

So, cube root of 110592 `=2xx2xx2xx2xx3=48`

(viii) 46656

**Answer:** `46656 = 2 xx 2 xx 2 xx 2 xx 2 xx 2 ``xx 3 xx 3 xx 3 xx 3 xx 3 xx 3`

`= 2^3 xx 2^3 xx 3^3 xx 3^3`

So, cube root of 46656 `=2xx2xx3xx3=36`

(ix) 175616

**Answer:** `175616 = 2^3 xx 2^3 xx 2^3 xx 7^3`

So, cube root of 175616 `=2xx2xx2xx7=56`

(x) 91125

**Answer:** `91125 = 5^3 xx 3^3 xx 3^3`

So, cube root of 91125 `=5xx3xx3=45`

Question 2: State true or false.

(i) Cube of any odd number is even.

**Answer:** FALSE: Odd multiplied by odd is always odd

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

**Answer:** TRUE: A perfect cube will end with odd number of zeroes

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

**Answer:** TRUE: 5 multiplied by 5 any number of times always gives 5 at unit’s place

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

**Answer:** False: `2^3= 8`

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

**Answer:** FALSE: The smallest two digit number is 10 and 10^{3 }= 1000 is a three digit number

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

**Answer:** FALSE: 99 is the largest 2 digit number; 99^{3 }= 989901 is a 6 digit number

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

**Answer:** TRUE: 2^{3 }= 8 is a single digit number

Question 3: You are told that 1,331 is a perfect cube. Can you guess without factorisation what is its cube root? Similarly, guess the cube root of 4913.

**Answer:** Let us divide 1331 in two groups of 31 and 13 for extreme right half and extreme left half of the number.

As you know 1^{3}= 1 so there would be 1 at unit’s place in cube root of 1331.

Now 2^{3}= 8 and 3^{3 }= 27

It is clear that 8 < 13 < 27, so the 10s digit of cube root of 1331 may be 2

So, cube root of 1331 may be 21 but 21^{3}= 9261 is not equal to 1331

So, let us test the 10s digit as 1

11^{3}= 1331 satisfies the condition

**4913:**

Right group = 13

Left group = 49

7^{3} gives 3 at unit’s place so unit digit number in cube root of 4913 should be 7

3^{3}= 27 and 4^{3 }= 64

27 < 49 < 64

So, 10s digit in cube root of 4913 should be 3

Test: 37^{3}= 50653 is not equal to 4913

Let us test 27^{3}= 19683 ≠ 4913 gives the answer

Let us test 17^{3}= 4913 gives the answer