Std 8 Mathematics · Chapter 3

3. Indices and Cube Root

Solutions to Practice Sets 3.1, 3.2 and 3.3, with factorisation steps. Maharashtra State Board (Balbharati), Standard Eight Mathematics.

Practice Set 3.1

1Express the following numbers in index form.

The nth root of a is written as a¹ⁿ.

(1) Fifth root of 1313¹⁵

(2) Sixth root of 99¹⁶

(3) Square root of 256256¹²

(4) Cube root of 1717¹³

(5) Eighth root of 100100¹⁸

(6) Seventh root of 3030¹⁷

In (3), 256¹² = 16, since 16 × 16 = 256. The question asks only for the index form.

2Write in the form ‘nth root of a’ in each of the following numbers.

(1) 81¹⁴Fourth root of 81

(2) 49¹²Square root of 49

(3) 15¹⁵Fifth root of 15

(4) 512¹⁹Ninth root of 512

(5) 100¹¹⁹Nineteenth root of 100

(6) 6¹⁷Seventh root of 6

The index 12 means square root, and 13 means cube root.

Practice Set 3.2

1Complete the following table.

For a^mn: power of the root = mth power of nth root of a; root of the power = nth root of mth power of a.

Sr. No.	Number	Power of the root	Root of the power
(1)	(225)³²	Cube of square root of 225	Square root of cube of 225
(2)	(45)⁴⁵	4th power of 5th root of 45	5th root of 4th power of 45
(3)	(81)⁶⁷	6th power of 7th root of 81	7th root of 6th power of 81
(4)	(100)⁴¹⁰	4th power of 10th root of 100	10th root of 4th power of 100
(5)	(21)³⁷	3rd power of 7th root of 21	7th root of 3rd power of 21

Row (1) is the example given in the textbook.

2Write the following numbers in the form of rational indices.

The power goes in the numerator of the index and the root goes in the denominator.

(1) Square root of 5th power of 121(121)⁵²

(2) Cube of 4th root of 324(324)³⁴

(3) 5th root of square of 264(264)²⁵

(4) Cube of cube root of 33³³ = 3

In (4), the index 33 equals 1, so 3³³ = 3¹ = 3. Taking the cube root and then cubing brings back the same number.

Practice Set 3.3

1Find the cube roots of the following numbers.

Write the prime factors, make groups of three equal factors, and take one factor from each group.

(1) 80008000 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5
= (2 × 2 × 5) × (2 × 2 × 5) × (2 × 2 × 5)
= 20³
∛8000 = 20

(2) 729729 = 3 × 3 × 3 × 3 × 3 × 3
= (3 × 3) × (3 × 3) × (3 × 3)
= 9³
∛729 = 9

(3) 343343 = 7 × 7 × 7 = 7³
∛343 = 7

(4) −512512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
= (2 × 2 × 2)³ = 8³
∴ −512 = (−8)³
∛−512 = −8

(5) −27442744 = 2 × 2 × 2 × 7 × 7 × 7
= (2 × 7)³ = 14³
∴ −2744 = (−14)³
∛−2744 = −14

(6) 3276832768 = 2¹⁵ (2 taken 15 times)
= (2⁵)³ = 32³
∛32768 = 32

The cube of a negative number is negative, so the cube root of a negative number is also negative.

2Simplify.

(1) ∛27125

∛27125 = ∛27∛125 = ∛3³∛5³ = 35

Answer: 35

(2) ∛1654

First simplify the fraction: 1654 = 16 ÷ 254 ÷ 2 = 827.

∛827 = ∛8∛27 = ∛2³∛3³ = 23

Answer: 23

In (2), 16 and 54 are not perfect cubes, so reduce the fraction first to get perfect cubes.

3If ∛729 = 9, then ∛0.000729 = ?

0.000729 has 6 decimal places, so 0.000729 = 7291000000 = 72910⁶.

∛0.000729 = ∛729∛10⁶ = 910² = 9100 = 0.09

Answer: ∛0.000729 = 0.09

Check: 0.09 × 0.09 × 0.09 = 0.000729. The cube has three times as many decimal places as the number (2 × 3 = 6).

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