R d sharma solutions mathematics class 8 Solutions for Chapter 4 cube and cube roots

Exercises in Chapter 4 cube and cube roots Grade 8

Cube and Cube Roots Exercise 4.1

Cube and Cube Roots Exercise 4.2

Cube and Cube Roots Exercise 4.3

Cube and Cube Roots Exercise 4.4

Cube and Cube Roots Exercise 4.5

Questions in Cube and Cube Roots Exercise 4.1

Find the cubes of the following numbers:

(i) 7

(ii) 12

(iii) 16

(iv) 21

(v) 40

(vi) 55

(vii) 100

(viii) 302

(ix) 301

Observe the following pattern:

13 = 1

13 + 23 = (1+2)2

13 + 23 + 33 = (1+2+3)2

Write the next three rows and calculate the value of 13 + 23 + 33 +…+ 93 by the above pattern.

Write the cubes of 5 natural numbers which are multiples of 3 and verify the followings:

“The cube of a natural number which is a multiple of 3 is a multiple of 27’

Write the cubes of 5 natural numbers which are of the form 3n + 1 (e.g. 4, 7, 10, …) and verify the following:

“The cube of a natural number of the form 3n+1 is a natural number of the same from i.e. when divided by 3 it leaves the remainder 1’

Write the cubes 5 natural numbers of the from 3n+2(i.e.5,8,11….) and verify the following:

“The cube of a natural number of the form 3n+2 is a natural number of the same form i.e. when it is dividend by 3 the remainder is 2’

Write the cubes of 5 natural numbers of which are multiples of 7 and verity the following:

“The cube of a multiple of 7 is a multiple of 73 .

Which of the following are perfect cubes?

(i) 64 (ii) 216

(iii) 243 (iv) 1000

(v) 1728 (vi) 3087

(vii) 4608 (viii) 106480

(ix) 166375 (x) 456533

Which of the following are cubes of even natural numbers?

216, 512, 729, 1000, 3375, 13824

Which of the following are cubes of odd natural numbers?

125, 343, 1728, 4096, 32768, 6859

What is the smallest number by which the following numbers mustbe multiplied, so that the products are perfect cubes?

(i) 675 (ii) 1323

(iii) 2560 (iv) 7803

(v) 107811 (vi) 35721

By which smallest number must the following numbers be divided so thatthe quotient is a perfect cube?

(i) 675 (ii) 8640

(iii) 1600 (iv) 8788

(v) 7803 (vi) 107811

(vii) 35721 (viii) 243000

Prove that if a number is trebled then its cube is 27 time the cube of the given number.

What happens to the cube of a number if the number is multiplied by

(i) 3?

(ii) 4?

(iii) 5?

Find the volume of a cube, one face of which has an area of 64m2 .

Find the volume of a cube whose surface area is 384m2 .

Evaluate the following:

(i) {(52 + 122)1/2}3

(ii) {(62 + 82)1/2}3

Write the units digit of the cube of each of the following numbers:

31, 109, 388, 4276, 5922, 77774, 44447, 125125125

Find the cubes of the following numbers by column method:

(i) 35

(ii) 56

(iii) 72

Which of the following numbers are not perfect cubes?

(i) 64

(ii) 216

(iii) 243

(iv) 1728

For each of the non-perfect cubes in Q. No 20 find the smallest number by which it must be

(a) Multiplied so that the product is a perfect cube.

(b) Divided so that the quotient is a perfect cube.

By taking three different, values of n verify the truth of the following statements:

(i) If n is even, then n3 is alsoeven.

(ii) If n is odd, then n3 is alsoodd.

(ii) If n leaves remainder 1 when divided by 3, then n3 also leaves 1 as remainder when divided by 3.

(iv) If a natural number n is of the form 3p+2 then n3 also a number of the same type.

Write true (T) or false (F) for the following statements:

(i) 392 is a perfect cube.

(ii) 8640 is not a perfect cube.

(iii) No cube can end with exactly two zeros.

(iv) There is no perfect cube which ends in 4.

(v) For an integer a, a3 is always greater than a2 .

(vi) If a and b are integers such that a2

b2 , then a3 b3 .

(vii) If a divides b, then a3 divides b3

(viii) If a2 ends in 9, then a3 ends in 7.

(ix) If a2 ends in an even number of zeros, then a3 ends in 25.

(x) If a2 ends in an even number of zeros, then a3 ends in an odd number of zeros.

Questions in Cube and Cube Roots Exercise 4.2

Find the cubes of:

(i) -11

(ii) -12

(iii) -21

Write the cubes of all natural numbers between 1 and 10 and verify the following statements:

(i) Cubes of all odd natural numbers are odd.

(ii) Cubes of all even natural numbers are even.

Which of the following integers are cubes of negative integers

(i) -64

(ii) -1056

(iii) -2197

(iv) -2744

(v) -42875

Show that the following integers are cubes of negative integers. Also, findthe integer whose cube is the given integer.

(i) -5832

(ii) -2744000

Find the cube of:

(i) 7/9 (ii) -8/11

(iii) 12/7 (iv) -13/8

(v) 2 2/5 (vi) 3 1/4

(vii) 0.3 (viii) 1.5

(ix) 0.08 (x) 2.1

Find which of the following numbers are cubes of rational numbers:

(i) 27/64

(ii) 125/128

(iii) 0.001331

(iv) 0.04

Questions in Cube and Cube Roots Exercise 4.3

Find the cube roots of the following numbers by successive subtractionof numbers:

1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, …

(i) 64

(ii) 512

(iii) 1728

Using the method of successive subtraction examine whether or not the following numbers are perfect cubes:

(i) 130

(ii) 345

(iii) 792

(iv) 1331

Find the smallest number that must be subtracted from those of the numbers in question 2 which are not perfect cubes, to make them perfect cubes. What are the corresponding cube roots?

Find the cube root of each of the following natural numbers:

(i) 343 (ii) 2744

(iii) 4913 (iv) 1728

(v) 35937 (vi) 17576

(vii) 134217728 (viii) 48228544

(ix) 74088000 (x) 157464

(xi) 1157625 (xii) 33698267

Find the smallest number which when multiplied with 3600 will make the product a perfect cube. Further, find the cube root of the product.

Multiply 210125 by the smallest number so that the product is a perfectcube. Also, find out the cube root of the product.

What is the smallest number by which 8192 must be divided sothat quotient is a perfect cube? Also, find the cube root of the quotient so obtained.

Three numbers are in the ratio 1:2:3. The sum of their cubes is 98784. Findthe numbers.

The volume of a cube is 9261000 m3 . Find the side of the cube.

Questions in Cube and Cube Roots Exercise 4.4

Find the cube roots of each of the following integers:

(i)-125 (ii) -5832

(iii)-2744000 (iv) -753571

(v) -32768

Show that:

(i) ∛27 × ∛64 = ∛ (27×64)

(ii) ∛ (64×729) = ∛64 × ∛729

(iii) ∛ (-125×216) = ∛-125 × ∛216

(iv) ∛ (-125×-1000) = ∛-125 × ∛-1000

Find the cube root of each of the following numbers:

(i) 8×125

(ii) -1728×216

(iii) -27×2744

(iv) -729×-15625

Evaluate:

(i) ∛ (43 × 63)

(ii) ∛ (8×17×17×17)

(iii) ∛ (700×2×49×5)

(iv) 125 ∛a6 – ∛125a6

Find the cube root of each of the following rational numbers:

(i) -125/729

(ii) 10648/12167

(iii) -19683/24389

(iv) 686/-3456

(v) -39304/-42875

Find the cube root of each of the following rational numbers:

(i) 0.001728

(ii) 0.003375

(iii) 0.001

(iv) 1.331

Evaluate each of the following:

(i) ∛27 + ∛0.008 + ∛0.064

(ii) ∛1000 + ∛0.008 – ∛0.125

(iii) ∛(729/216) × 6/9

(iv) ∛(0.027/0.008) ÷ ∛(0.09/0.04) – 1

(v) ∛(0.1×0.1×0.1×13×13×13)

Show that:

(i) ∛ (729)/ ∛ (1000) = ∛ (729/1000)

(ii) ∛ (-512)/ ∛ (343) = ∛ (-512/343)

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

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

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

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

Evaluate:

(i) ∛36 × ∛384

(ii) ∛96 × ∛144

(iii) ∛100 × ∛270

(iv) ∛121 × ∛297

Find the cube roots of the numbers 3048625, 20346417, 210644875, 57066625 using the fact that

(i) 3048625 = 3375 × 729

(ii) 20346417 = 9261 × 2197

(iii) 210644875 = 42875 × 4913

(iv) 57066625 = 166375 × 343

Find the unit of the cube root of the following numbers:

(i) 226981

(ii) 13824

(iii) 571787

(iv) 175616

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

Questions in Cube and Cube Roots Exercise 4.5

Making use of the cube root table, find the cube root of the following (correct to three decimal places): 7