Mathematics

Cube Root – Ch.2 – cubes and cube roots – Class VIII

If N is number and n is another number such that N = n3, we say n is the cube root of N and write n =  cube root of N .

i.e.,

Examples for Cube root

Example 12: Find the cube root of 216 by factorization.

Solution:

216 = 2 x (108) = 2 x 2 x 54 = 2 x 2 x 2 x 27 = 2 x 2 x 2 x 3 x 9 = 2 x 2 x 2 x 3 x 3 x 3

216 = (2 x 3) x (2 x 3) x (2 x 3)

216  = 6 x 6 x 6 = 63

Example 13: find the cube root of -17576 using factorization.

Solution:

-17576 = 2 x (-8788) = 2 x 2 x (-4394) = 2 x 2 x 2 x (-2197) = 2 x 2 x 2 x (-13) x (-13) x (-13)

= [2 x (-13)] x [2 x (-13)] x [2 x (-13)]

= (-26) x (-26) x (-26)

= (-26)3

Example 15: Find the cube root of 103823.

Solution:

Here unit in the digits place is 3. If n3 = 103823. Then, the unit in the digits place of n must be 7.

Let us split this as 103 and 823.

We observe that, 43 = 64 < 103 < 125 = 53.

Hence 403 = 64000 < 103823 < 125000 = 503. Hence n must lie between 40 and 50. Since the unit in the digits place is 7, therefore n must be 47.

473 = 103823.

Cube root – Chapter 2 – Exercise 1.2.8

1. Find the cube root by prime factorization.

i) 10648

Solution:

10648 =  2 x 5324

= 2 x 2 x 2662

= 2 x 2 x 2 x 1331

= 2 x 2 x 2 x 11 x 121

= 2 x 2 x 2 x 11 x 11 x 11

= (2 x 11) x (2 x 11) x (2 x 11)

= 22 x 22 x 22

= 223.

ii) 46656

Solution:

46656 = 2 x (23328)

= 2 x 2x (11664)

= 2 x 2 x 2 x (5832)

= 2x2x2x2x(2916)

= 2x2x2x2x2x(1458)

= 2x2x2x2x2x2x(729)

= 2x2x2x2x2x2x9x(81) = 2x2x2x2x2x2x9x9x9 = (2x2x9)x(2x2x9)x(2x2x9) = 36 x 36 x36

= 363.

iii)15625

Solution:

15625 = 5 x (3125)

= 5 x 5 x (625)

= 5 x 5 x 5 x (125)

= 5 x 5 x 5 x 5 x (25)

= 5 x 5 x 5 x 5 x 5 x 5

= (5×5)x(5×5)x(5×5)

= (25) x (25) x (25)

= 253

1. Find the cube root of the following by looking at the last digit and using estimation.

i) 91125

Solution:

Here unit in the digits place is 5. If n3 = 91125. Then, the unit in the digits place of n must be 5.

Let us split this as 91 and 125.

We observe that, 43 = 64 < 91 < 125 = 53.

Hence 403 = 64000 < 91125 < 125000 = 503. Hence n must lie between 40 and 50. Since the unit in the digits place is 5, therefore n must be 45.

ii) 166375

Solution:

Here unit in the digits place is 5. If n3 = 166375. Then, the unit in the digits place of n must be 5.

Let us split this as 166 and 375.

We observe that, 53 = 125 < 166 < 216 = 63.

Hence 503 = 12500 < 166375 < 216000 = 603. Hence n must lie between 50 and 60. Since the unit in the digits place is 5, therefore n must be 55.

iii) 704969

Solution:

Here unit in the digits place is 9. If n3 = 704969. Then, the unit in the digits place of n must be 9.

Let us split this as 704 and 969.

We observe that, 83 = 512 < 704 < 729 = 93.

Hence 803 = 512000 < 704969 < 729000 = 903. Hence n must lie between 80 and 90. Since the unit in the digits place is 9, therefore n must be 89.

1. Find the nearest integer to the cube root of each of the following.

i) 331776

Solution:

For easy simplification let us split 331776 as 331 and 776,

63 = 216 < 331 < 343 = 73

603 = 216000 < 331776 < 343000 = 703.

Now it is closer to 703 than 603.

Let us go for more accurate, 693 = 328509 < 331776 < 34300 = 703.

Therefore , 331776 is closest to 693.

ii) 46656

Solution:

For easy simplification let us split 46656 as 46 and 656,

33 = 27 < 46 < 64 = 43

303 = 27000 < 46656 < 64000 = 403.

Now it is closer to 403 than 303.

The number closer to 40 than 30 are 36, 37, 38, 39. Let us go through one by one.

363 = 46656, satisfies the condition.

iii. 373248

Solution:

For easy simplification let us split 372248 as 373 and 248,

73 = 343 < 373 < 512 = 83

703 = 343000 < 373248 < 512000 = 803.

Now, it is closer to 703 than 803.

The numbers which are closer to 70 than 80 are : 71, 72, 73, 74, 75

Let us go for more accurate, 713 = 357911 < 373248 < 373248 = 723.

Therefore, 373248 is closest to 723

Squares, Square roots, Cubes and Cube roots

2 thoughts on “Cube Root – Ch.2 – cubes and cube roots – Class VIII”

1. It’s been more than a decade since my last Math test, but I still have nightmares about it 🙂

The Curious Incident of the Dog in the Night-time – the must book for you to read. I’ve always been fascinated by people who are good with numbers.

Liked by 1 person