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
= 23 × 23
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 103 = 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; 993 = 989901 is a 6 digit number
(vii) The cube of a single digit number may be a single digit number.
Answer: TRUE: 23 = 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 13= 1 so there would be 1 at unit’s place in cube root of 1331.
Now 23= 8 and 33 = 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 213= 9261 is not equal to 1331
So, let us test the 10s digit as 1
113= 1331 satisfies the condition
4913:
Right group = 13
Left group = 49
73 gives 3 at unit’s place so unit digit number in cube root of 4913 should be 7
33= 27 and 43 = 64
27 < 49 < 64
So, 10s digit in cube root of 4913 should be 3
Test: 373= 50653 is not equal to 4913
Let us test 273= 19683 ≠ 4913 gives the answer
Let us test 173= 4913 gives the answer