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

So, cube root of 64 `=2xx2=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



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