**If N is number and n is another number such that N = n ^{3}, we say n is the cube root of N and write n = **

**.**

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 = 6^{3}

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 n^{3} = 103823. Then, the unit in the digits place of n must be 7.

Let us split this as 103 and 823.

We observe that, 4^{3 }= 64 < 103 < 125 = 5^{3}.

Hence 40^{3} = 64000 < 103823 < 125000 = 50^{3}. Hence n must lie between 40 and 50. Since the unit in the digits place is 7, therefore n must be 47.

47^{3} = 103823.

**Exercise 1.2.8**

- 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

= 22^{3}.

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

= 36^{3}.

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)

= 25^{3}

- 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 n^{3} = 91125. Then, the unit in the digits place of n must be 5.

Let us split this as 91 and 125.

We observe that, 4^{3 }= 64 < 91 < 125 = 5^{3}.

Hence 40^{3} = 64000 < 91125 < 125000 = 50^{3}. 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 n^{3} = 166375. Then, the unit in the digits place of n must be 5.

Let us split this as 166 and 375.

We observe that, 5^{3 }= 125 < 166 < 216 = 6^{3}.

Hence 50^{3} = 12500 < 166375 < 216000 = 60^{3}. 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 n^{3} = 704969. Then, the unit in the digits place of n must be 9.

Let us split this as 704 and 969.

We observe that, 8^{3 }= 512 < 704 < 729 = 9^{3}.

Hence 80^{3} = 512000 < 704969 < 729000 = 90^{3}. Hence n must lie between 80 and 90. Since the unit in the digits place is 9, therefore n must be 89.

- 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,

6^{3 }= 216 < 331 < 343 = 7^{3}

60^{3} = 216000 < 331776 < 343000 = 70^{3}.

Now it is closer to 70^{3 }than 60^{3}.

Let us go for more accurate, 69^{3 }= 328509 < 331776 < 34300 = 70^{3}.

Therefore , 331776 is closest to 69^{3}.

ii46656

**Solution:**

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

3^{3 }= 27 < 46 < 64 = 4^{3}

30^{3} = 27000 < 46656 < 64000 = 40^{3}.

Now it is closer to 40^{3 }than 30^{3}.

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

36^{3 }= 46656, satisfies the condition.

iii. 373248

**Solution:**

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

7^{3 }= 343 < 373 < 512 = 8^{3}

70^{3} = 343000 < 373248 < 512000 = 80^{3}.

Now, it is closer to 70^{3 }than 80^{3}.

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

Let us go for more accurate, 71^{3 }= 357911 < 373248 < 373248 = 72^{3}.

Therefore, 373248 is closest to 72^{3}

## 2 responses to “1.2.8 Cube root”

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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.

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