Class 8 RD Sharma Solution – Chapter 4 Cubes and Cube Roots – Exercise 4.4 | Set 2

Question 9. Fill in the blanks:

(i) ∛(125 × 27) = 3 × …

(ii) ∛(8 × …) = 8

(iii) ∛1728 = 4 × …

(iv) ∛480 = ∛3 × 2 × ∛..

(v) ∛… = ∛7 × ∛8

(vi) ∛..= ∛4 × ∛5 × ∛6

(vii) ∛(27/125) = …/5

(viii) ∛(729/1331) = 9/…

(ix) ∛(512/…) = 8/13

Solution:

(i) ∛(125 × 27) = 3 × ….

Let’s solve LHS,

 ∛(125 × 27) = ∛(5 × 5 × 5 × 3 × 3 × 3) 

= 5 × 3

So missing value in RHS is 5

(ii) ∛(8 × …) = 8

Let’s solve LHS,

 ∛(8 × …) = ∛8 × 8 × 8 = 8

So missing value in LHS is 8

(iii) ∛1728 = 4 × …

Let’s solve LHS,

 ∛1728 = ∛2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3

= 2 × 2 × 3 = 4 × 3

So missing value in RHS is 3

(iv) ∛480 = ∛3×2× ∛..

Let’s solve LHS,

 ∛480 = ∛2 × 2 × 2 × 2 × 2 × 3 × 5 = 2 × ∛3 × ∛2× 2 × 5 

= 2 × ∛3 × ∛20

So missing value in RHS is 20

(v) ∛… = ∛7 × ∛8

Let’s solve RHS,

∛7 × ∛8 = ∛(7 × 8) = ∛56

So missing value in LHS is 56

(vi) ∛..= ∛4 × ∛5 × ∛6

Let’s solve RHS,

∛4 × ∛5 × ∛6 = ∛(4 × 5 × 6) = ∛120

So missing value in LHS is 120

(vii) ∛(27/125) = …/5

Let’s solve LHS,

∛(27/125)= ∛(3 × 3 × 3)/(5 × 5 × 5) = 3/5 

So missing value in RHS is 3

(viii) ∛(729/1331) = 9/…

Let’s solve LHS,

∛(729/1331)= ∛(9 × 9 × 9)/(11 × 11 × 11) = 9/11 

So missing value in RHS is 11

(ix) ∛(512/…) = 8/13

Let’s solve RHS,

Taking cube and cube root of 8/13 at the same time

So, ∛(8/13)³ =  ∛(8 × 8 × 8)/(13 × 13 × 13) = ∛512/2197

So missing value in LHS is 2197

Question 10. The volume of a cubical box is 474. 552 cubic meters. Find the length of each side of the box.

Solution:

Let a be the length of each side of the box

The volume of cubical box = 474.552 m³

As, we know volume of cube = (side)³

So, a³ = 474552/1000

a = ∛474552/1000 

Solving the cube root of 474552/1000

= ∛(2 × 2 × 2 × 3 × 3 × 3 × 13 × 13 × 13)/(10 × 10 × 10)

= 2 × 3 × 13/10 = 7.8 m

Hence, the length of each side of the box is 7.8 m

Question 11. Three numbers are to one another 2:3:4. The sum of their cubes is 0.334125. Find the numbers

Solution:

Let the three numbers be 2a, 3a, and 4a

It is given that the sum of their cubes is 0.334125

So, 2a³ + 3a³ + 4a³ = 334125/1000000

99a³ = 334125/1000000 or a = ∛334125/(1000000 × 99)

= ∛3375/1000000 = ∛15 × 15 × 15/10 × 10 × 10 × 10 × 10 × 10

= 15/100 = 0.15

So, first number = 2a = 2 × 0.15 = 0.3

The second number = 3a = 3 × 0.15 = 0.45

The third number = 4a = 4 × 0.15 = 0.6

Hence, the numbers are 0.3, 0.45, and 0.6

Question 12. Find the side of a cube whose volume is 24389/216m³.

Solution:

Let a be the length of each side of the cube

The volume of cube = 24389/216 m³

As, we know volume of cube = (side)³

So, a³ = 24389/216

a = ∛24389/216 

Solving the cube root of 24389/216

= ∛(29 × 29 × 29)/(2 × 2 × 2 × 3 × 3 × 3)

= 29/2 × 3 = 29/6 = 4.84 m

Hence, the length of each side of the cube is 4.84 m

Question 13. Evaluate:

(i) ∛36 × ∛384

(ii) ∛96 × ∛144

(iii) ∛100 × ∛270

(iv) ∛121 × ∛297

Solution :

(i) ∛36 × ∛384

We know that, ∛a × ∛b = ∛ab

= ∛(36 × 384) = ∛(2 × 2 × 3 × 3) × (2 × 2 × 2 × 2 × 2 × 2 × 2 × 3)

= ∛2³ × 2³ × 2³ × 3³

= 2 × 2 × 2 × 3 = 24

Hence, ∛36 × ∛384 = 24

(ii) ∛96 × ∛144

We know that, ∛a × ∛b = ∛ab

= ∛(96 × 144) = ∛(2 × 2 × 2 × 2 × 2 × 3) × (2 × 2 × 2 × 2 × 3 × 3)

= ∛2³ × 2³ × 2³ × 3³

= 2 × 2 × 2 × 3 = 24

Hence, ∛96 × ∛144 = 24

(iii) ∛100 × ∛270

We know that, ∛a × ∛b = ∛ab

= ∛(100 × 270) = ∛27000

= ∛3 × 3 × 3 × 10 × 10 × 10 = 3 × 10 = 30

Hence, ∛100 × ∛270 = 30

(iv) ∛121 × ∛297

We know that, ∛a × ∛b = ∛ab

= ∛(121 × 297) = ∛(11 × 11) × (3 × 3 × 3 × 11)

= ∛3³ × 11³

= 3 × 11 = 33

Hence, ∛121 × ∛297 = 33

Question 14. Find the cube roots of the numbers 2460375, 20346417, 210644875, 57066625 using the fact that

(i) 2460375 = 3375 × 729

(ii) 20346417 = 9261 × 2197

(iii) 210644875 = 42875 × 4913

(iv) 57066625 = 166375 × 343

Solution:

(i) 2460375 = 3375 × 729

Cube root of 2460375 will be written as,

∛2460375 = ∛3375 × 729

= ∛(3 × 3 × 3 × 5 × 5 × 5) × (9 × 9 × 9)

= ∛3³ × 5³ × 9³

= 3 × 5 × 9 = 135

Hence, cube root of 2460375 is 135

(ii) 20346417 = 9261 × 2197  

Cube root of 20346417 will be written as,

∛20346417 = ∛9261 × 2197  

= ∛(3 × 3 × 3 × 7 × 7 × 7) × (13 × 13 × 13)

= ∛3³ × 7³ × 13³

= 3 × 7 × 13 = 273

Hence, cube root of 20346417 is 273

(iii) 210644875 = 42875 × 4913 

Cube root of 210644875 will be written as,

∛210644875 = ∛42875 × 4913 

= ∛(5 × 5 × 5 × 7 × 7 × 7) × (17 × 17 × 17)

= ∛5³ × 7³ × 17³

= 5 × 7 × 17 = 595

Hence, cube root of 210644875 is 595

(iv) 57066625 = 166375 × 343

Cube root of 57066625 will be written as,

∛57066625 = ∛166375 × 343

= ∛(5 × 5 × 5 × 11 × 11 × 11) × (7 × 7 × 7)

= ∛5³ × 11³ × 7³

= 5 × 11 × 7 = 385

Hence, cube root of 57066625 is 385

Question 15. Find the unit of the cube root of the following numbers:

(i) 226981

(ii) 13824

(iii) 571787

(iv) 175616

Solution:

(i) 226981 

Since the unit digit of the given number is 1

So, the unit digit of cube root of 226981 is 1

(ii) 13824

Since the unit digit of the given number is 4

So, the unit digit of cube root of 13824 is 4

(iii) 571787

Since the unit digit of the given number is 7

So, the unit digit of cube root of 571787 is 7

(iv) 175616

Since the unit digit of the given number is 6

So, the unit digit of cube root of 175616 is 6

Question 16. Find the tens digit of the cube root of each of the numbers in Q.No.15.

(i) 226981

(ii) 13824

(iii) 571787

(iv) 175616

Solution:

(i) 226981 

Since the unit digit of the given number is 1

So, the unit digit of cube root of 226981 is 1

Let’s remove the unit, tens, and hundreds digit of the given number, we get 226 

As we know the number 226 lies between cube root of 6 and 7 (6³ < 226 < 7³)

So, 6 is the largest number whose cube root will be less than or equal to 226

Hence, the tens digit of the cube root of 226981 is 6

(ii) 13824

Since the unit digit of the given number is 4

So, the unit digit of cube root of 13824 is 4

Let’s remove the unit, tens, and hundreds digit of the given number, we get 13

As we know the number 13 lies between cube root of 2 and 3 (2³ < 13 < 3³)

So, 2 is the largest number whose cube root will be less than or equal to 13

Hence, the tens digit of the cube root of 13824 is 2

(iii) 571787

Since the unit digit of the given number is 7

So, the unit digit of cube root of 571787 is 3

Let’s remove the unit, tens, and hundreds digit of the given number, we get 571

As we know the number 571 lies between cube root of 8 and 9 (8³ < 571 < 9³)

So, 8 is the largest number whose cube root will be less than or equal to 571

Hence, the tens digit of the cube root of 571787 is 8

(iv) 175616

Since the unit digit of the given number is 6

So, the unit digit of cube root of 175616 is 6

Let’s remove the unit, tens, and hundreds digit of the given number, we get 175

As we know the number 175 lies between cube root of 5 and 6 (5³ < 571 < 6³)

So, 5 is the largest number whose cube root will be less than or equal to 175

Hence, the tens digit of the cube root of 175616 is 5