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

---

2. 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.

---

3. 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

---