CUBES AND CUBE-ROOTS - Q&A

EXERCISE 4(A)

1. Multiple Choice Type: Choose the correct answer from the options given below.

(i) The cube of 0.5 is:
(a) 1.25
(b) 0.125
(c) 12.5
(d) 0.0125
Answer: (b) 0.125
Steps: (0.5)³ = 0.5 × 0.5 × 0.5 = 0.125

(ii) The cube of -4 is:
(a) 16
(b) 12
(c) -64
(d) 64
Answer: (c) -64
Steps: (-4)³ = -4 × -4 × -4 = 16 × -4 = -64

(iii) The smallest number by which 72 must be multiplied to obtain a perfect cube is:
(a) 3
(b) 2
(c) 4
(d) 6
Answer: (a) 3
Steps: Prime factorization of 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3².
To make it a perfect cube, the power of 3 must be a multiple of 3. We have 3², so we need one more 3.

(iv) The smallest number by which 81 be divided to obtain a perfect cube is:
(a) 1
(b) 3
(c) 9
(d) none of the above
Answer: (b) 3
Steps: Prime factorization of 81 = 3 × 3 × 3 × 3 = 3⁴.
To make it a perfect cube, we remove the extra factor. 3³ is a cube. The extra factor is 3.

2. Find the cube of:

(i) 7
Answer: 343
Steps: 7 × 7 × 7 = 343

(ii) 11
Answer: 1331
Steps: 11 × 11 × 11 = 1331

(iii) 16
Answer: 4096
Steps: 16 × 16 × 16 = 4096

(iv) 23
Answer: 12167
Steps: 23 × 23 × 23 = 12167

(v) 31
Answer: 29791
Steps: 31 × 31 × 31 = 29791

(vi) 42
Answer: 74088
Steps: 42 × 42 × 42 = 74088

(vii) 54
Answer: 157464
Steps: 54 × 54 × 54 = 157464

3. Find which of the following are perfect cubes?

(i) 243 (ii) 588 (iii) 1331 (iv) 24000 (v) 1728 (vi) 1938
Answer: (iii) 1331 and (v) 1728
Steps:
(i) 243 = 3⁵ (Not a multiple of 3, not a perfect cube)
(ii) 588 = 2² × 3 × 7² (Not a perfect cube)
(iii) 1331 = 11³ (Perfect cube)
(iv) 24000 = 24 × 1000. 1000 is a cube, but 24 is not. (Not a perfect cube)
(v) 1728 = 12³ (Perfect cube)
(vi) 1938 = 2 × 3 × 17 × 19 (Not a perfect cube)

4. Find the cubes of:

(i) -3
Answer: -27
Steps: (-3)³ = -27

(ii) -7
Answer: -343
Steps: (-7)³ = -343

(iii) -12
Answer: -1728
Steps: (-12)³ = -1728

(iv) -18
Answer: -5832
Steps: (-18)³ = -5832

(v) -25
Answer: -15625
Steps: (-25)³ = -15625

(vi) -30
Answer: -27000
Steps: (-30)³ = -27000

(vii) -50
Answer: -125000
Steps: (-50)³ = -125000

5. Find the cubes of:

(i) 2 1/2
Answer: 15 5/8
Steps: 2 1/2 = 5/2. (5/2)³ = 125/8 = 15 5/8.

(ii) 3/7
Answer: 27/343
Steps: (3/7)³ = 27/343.

(iii) 1 2/7
Answer: 2 10/343
Steps: 1 2/7 = 9/7. (9/7)³ = 729/343 = 2 43/343.

(iv) 8/9
Answer: 512/729
Steps: (8/9)³ = 512/729.

(v) 10/13
Answer: 1000/2197
Steps: (10/13)³ = 1000/2197.

6. Find the cubes of:

(i) 2.1
Answer: 9.261
Steps: (2.1)³ = 2.1 × 2.1 × 2.1 = 9.261

(ii) 0.4
Answer: 0.064
Steps: (0.4)³ = 0.064

(iii) 1.6
Answer: 4.096
Steps: (1.6)³ = 4.096

(iv) 2.5
Answer: 15.625
Steps: (2.5)³ = 15.625

(v) 0.12
Answer: 0.001728
Steps: (0.12)³ = 0.001728

(vi) 0.02
Answer: 0.000008
Steps: (0.02)³ = 0.000008

(vii) 0.8
Answer: 0.512
Steps: (0.8)³ = 0.512

7. Which of the following are cubes of: (i) an even number (ii) an odd number.
List: 216, 729, 3375, 8000, 125, 343, 4096 and 9261.
Answer:
(i) Even number cubes: 216, 8000, 4096.
(ii) Odd number cubes: 729, 3375, 125, 343, 9261.
Explanation: The cube of an even number is even, and the cube of an odd number is odd.

8. Find the least number by which 1323 must be multiplied so that the product is a perfect cube.
Answer: 7
Steps: Prime factorization of 1323 = 3 × 3 × 3 × 7 × 7 = 3³ × 7².
The factor 3 is in a triplet, but 7 is only a pair. We need one more 7 to make a triplet.

9. Find the smallest number by which 8768 must be divided so that the quotient is a perfect cube.
Answer: 137
Steps: Prime factorization of 8768 = 2 × 2 × 2 × 2 × 2 × 2 × 137 = 2⁶ × 137.
2⁶ is a perfect cube [(2²)³]. The extra factor is 137. Dividing by 137 removes it.

10. Find the smallest number by which 27783 should be multiplied to get a perfect cube number.
Answer: 9
Steps: 27783 = 3 × 9261 = 3 × (21)³ = 3 × (3 × 7)³ = 3 × 3³ × 7³.
Here 3 is the only single factor (ignoring the already perfect cube 3³). Wait, let's look at total powers: 3¹ × 3³ = 3⁴. We need the power of 3 to be a multiple of 3 (next is 6). We have 4, we need 2 more. So we multiply by 3 × 3 = 9.

11. With what least number should 8640 be divided so that the quotient is a perfect cube?
Answer: 5
Steps: 8640 = 10 × 864 = 2 × 5 × 2 × 432 = 2² × 5 × 2 × 216 = 2³ × 5 × 6³.
Factors: 2³ (cube) × 5 (single) × 6³ (cube). To get a perfect cube, remove the single factor 5.

12. Which is the smallest number that should be multiplied to 77175 to make it a perfect cube?
Answer: 15
Steps: 77175 = 25 × 3087 = 5² × 3087.
3087 is divisible by 3: 3087 = 3 × 1029 = 3 × 3 × 343 = 3² × 7³.
Total factorization: 5² × 3² × 7³.
We need triplets. We have two 5s (need one more). We have two 3s (need one more). 7s are already a triplet.
Multiply by 5 × 3 = 15.

EXERCISE 4(B)

1. Multiple Choice Type: Choose the correct answer from the options given below.

(i) The cube root of 0.000027 is:
(a) 0.03
(b) 0.003
(c) 0.3
(d) 0.00003
Answer: (a) 0.03
Steps: √[3]{27/1000000} = 3/100 = 0.03

(ii) The cube root of -0.064 is:
(a) -0.8
(b) 0.8
(c) 0.4
(d) -0.4
Answer: (d) -0.4
Steps: √[3]{-64/1000} = -4/10 = -0.4

2. Find the cube-roots of:

(i) 64 Answer: 4

(ii) 343 Answer: 7

(iii) 729 Answer: 9

(iv) 1728 Answer: 12

(v) 9261 Answer: 21

(vi) 4096 Answer: 16

(vii) 8000 Answer: 20

(viii) 3375 Answer: 15

3. Find the cube-roots of:

(i) 27/64
Answer: 3/4
Steps: √[3]{27}/√[3]{64} = 3/4

(ii) 125/216
Answer: 5/6
Steps: √[3]{125}/√[3]{216} = 5/6

(iii) 343/512
Answer: 7/8
Steps: √[3]{343}/√[3]{512} = 7/8

(iv) 64 × 729
Answer: 36
Steps: √[3]{64} × √[3]{729} = 4 × 9 = 36

(v) 64 × 27
Answer: 12
Steps: √[3]{64} × √[3]{27} = 4 × 3 = 12

(vi) 729 × 8000
Answer: 180
Steps: √[3]{729} × √[3]{8000} = 9 × 20 = 180

(vii) 3375 × 512
Answer: 120
Steps: √[3]{3375} × √[3]{512} = 15 × 8 = 120

4. Find the cube-roots of:

(i) -216 Answer: -6

(ii) -512 Answer: -8

(iii) -1331 Answer: -11

(iv) -27/125 Answer: -3/5

(v) -64/343 Answer: -4/7

(vi) -512/343 Answer: -8/7

(vii) -2197 Answer: -13

(viii) -5832 Answer: -18

(ix) -2744000 Answer: -140 (Steps: √[3]{-2744} × √[3]{1000} = -14 × 10)

5. Find the cube-roots of:

(i) 2.744 Answer: 1.4 (Steps: √[3]{2744/1000} = 14/10)

(ii) 9.261 Answer: 2.1 (Steps: √[3]{9261/1000} = 21/10)

(iii) 0.000027 Answer: 0.03

(iv) -0.512 Answer: -0.8

(v) -15.625 Answer: -2.5 (Steps: √[3]{-15625/1000} = -25/10)

(vi) -125 × 1000 Answer: -50 (Steps: -5 × 10)

6. Find the smallest number by which 26244 should be divided so that the quotient is a perfect cube.
Answer: 36
Steps: 26244 = 4 × 6561 = 2² × 9 × 729 = 2² × 3² × 9³.
To simplify: 26244 = 2² × 3² × (3²)³ = 2² × 3² × 3⁶.
3⁶ is a perfect cube. The extra factors are 2² × 3² = 4 × 9 = 36.
Divide by 36. Quotient = 729 (which is 9³).

7. What is the least number by which 30375 should be multiplied to get a perfect cube?
Answer: 3
Steps: 30375 = 5 × 6075 = 5 × 25 × 243 = 5³ × 3⁵.
We have 5³ (perfect cube) and 3⁵.
To make 3⁵ a perfect cube (closest is 3⁶), we need one more 3.
Multiply by 3.

8. Find the cube-roots of:

(i) 700 × 2 × 49 × 5
Answer: 70
Steps: 700 × 2 × 49 × 5 = (7 × 100) × 2 × 7² × 5 = 7³ × (100 × 2 × 5) = 7³ × 1000 = 7³ × 10³.
Cube root = 7 × 10 = 70.

(ii) -216 × 1728
Answer: -72
Steps: √[3]{-216} × √[3]{1728} = -6 × 12 = -72.

(iii) -64 × -125
Answer: 20
Steps: √[3]{-64} × √[3]{-125} = -4 × -5 = 20.

(iv) -27/343
Answer: -3/7

(v) 729 / -1331
Answer: -9/11

(vi) 250.047
Answer: 6.3
Steps: 250047/1000 = (27 × 9261) / 1000 = (3³ × 21³) / 10³.
Root = (3 × 21) / 10 = 63/10 = 6.3.

(vii) -175616
Answer: -56
Steps: 175616 ends in 6, so root ends in 6. 175 is between 125 (5³) and 216 (6³). So it is 56.

Test yourself

1. Multiple Choice Type: Choose the correct answer from the options given below.

(i) 64x³ - √[64x⁶] is equal to:
(a) 72x³
(b) 56x²
(c) 128x³
(d) 56x³
Answer: (d) 56x³
Steps: √[64x⁶] = 8x³ (Square root of x⁶ is x³).
Expression: 64x³ - 8x³ = 56x³.

(ii) If a number is multiplied by 3, its square will be multiplied by:
(a) 9
(b) 3
(c) 27
(d) 81
Answer: (a) 9
Steps: (3n)² = 9n².

(iii) Two numbers are in the ratio 5:4. If the difference of their cubes is 61; the numbers are:
(a) 5 and 4
(b) 25 and 16
(c) 10 and 18
(d) none of these
Answer: (a) 5 and 4
Steps: Let numbers be 5x and 4x.
(5x)³ - (4x)³ = 61
125x³ - 64x³ = 61
61x³ = 61 ⇒ x³ = 1 ⇒ x = 1.
Numbers are 5(1) and 4(1).

(iv) The value of √[3]{27} + √[3]{0.008} + √[3]{0.064} is:
(a) √[3]{27.072}
(b) 3.72
(c) 3.6
(d) 3.06
Answer: (c) 3.6
Steps: 3 + 0.2 + 0.4 = 3.6

(v) The value of √[3]{(-3)³ × 8} is:
(a) -27
(b) -6
(c) 6
(d) none of these
Answer: (b) -6
Steps: √[3]{-27 × 8} = √[3]{-216} = -6.

(vi)

Statement 1: Cubes of all odd natural numbers are odd.
Statement 2: Cubes of negative integers are positive or negative integers.
(a) Both the statements are true.
(b) Both the statements are false.
(c) Statement 1 is true, and statement 2 is false.
(d) Statement 1 is false, and statement 2 is true.
Answer: (a) Both the statements are true.

(vii) Assertion (A): The smallest number by which 1323 may be multiplied so that the product is a perfect cube is 7.

Reason (R): A given natural number is a perfect cube if in its prime factorization every prime occurs three times.
(a) (1) Both A and R are correct and R is correct explanation.
(b) (2)
(c) (3)
(d) (4)
Answer: (a) (1)
Steps: 1323 = 3³ × 7². Need one 7. So A is correct. R is the definition of perfect cube.

(viii) Assertion (A): √[3]{4 12/125} = 1 3/5
Reason (R): √[3]{p/q} = √[3]{p}/√[3]{q}
Answer: (a) (1)
Steps: 4 12/125 = (500+12)/125 = 512/125. √[3]{512}/√[3]{125} = 8/5 = 1 3/5. Correct.

(ix) Assertion (A): √[3]{-125} = ±25
Reason (R): The cube root of a negative perfect cube is negative.
Answer: (d) (4) A is false, but R is true.
Steps: √[3]{-125} = -5, not ±25.

(x) Assertion (A): √[3]{968} × √[3]{1375} = 110
Reason (R): √[3]{p} × √[3]{q} = √[3]{pq}
Answer: (a) (1)
Steps: √[3]{968 × 1375} = √[3]{(8 × 121) × (125 × 11)} = √[3]{8 × 125 × 1331} = 2 × 5 × 11 = 110.

2. State true or false:

(i) Cube of an odd number can be even. False

(ii) A perfect cube does not end with two zeroes. True (Must end in 3 zeroes or multiples of 3)

(iii) If square of a number ends with 5, its cube will end with 25. True (Number ends in 5 -> Square ends in 25 -> Cube ends in 125, which ends in 25)

(iv) The cube of a two digit number may be a three digit number. False (Smallest 2-digit is 10, 10³=1000, which is 4-digit)

(v) Cube of a natural number is called perfect cube. True

3. Find the cube roots of:

(i) 110.592
Answer: 4.8
Steps: 110592 ends in 2, so root ends in 8. 110 is between 64 (4³) and 125 (5³). So 48. Divide by 10.

(ii) 0.064
Answer: 0.4

4. Find the volume of a cubical box whose surface area is 486 cm².
Answer: 729 cm³
Steps: Surface Area = 6a² = 486.
a² = 486/6 = 81.
a = 9 cm.
Volume = a³ = 9³ = 729 cm³.

5. Find cube roots of:

(i) 125 × -64
Answer: -20
Steps: 5 × -4 = -20

(ii) -125/343
Answer: -5/7

6. Three numbers are in the ratio 2:3:1. The sum of their cubes is 288. Find the numbers.
Answer: 4, 6, 2
Steps: Let numbers be 2x, 3x, x.
(2x)³ + (3x)³ + (x)³ = 288
8x³ + 27x³ + x³ = 288
36x³ = 288
x³ = 8 ⇒ x = 2.
Numbers: 2(2)=4, 3(2)=6, 1(2)=2.

7. Find the smallest number by which 14,580 must be multiplied to make a perfect cube. Also, find the cube root of the perfect cube number obtained.
Answer: Multiply by 50; Cube root is 90.
Steps: 14580 = 10 × 1458 = 2 × 5 × 2 × 729 = 2² × 5 × 9³.
We have 2² (need one 2) and 5¹ (need two 5s).
Multiply by 2 × 5 × 5 = 50.
New number = 14580 × 50 = 729000.
Cube root = √[3]{729000} = 90.

8. Find the smallest number by which 8,232 must be divided to make it a perfect cube. Also, find the cube root of the perfect cube so obtained.
Answer: Divide by 3; Cube root is 14.
Steps: 8232 = 8 × 1029 = 2³ × 3 × 343 = 2³ × 3 × 7³.
The extra factor is 3.
Divide by 3 to get 2³ × 7³.
Root = 2 × 7 = 14.

9. Evaluate:

(i) [(12² + 5²)^1/2]³
Answer: 2197
Steps: 12² + 5² = 144 + 25 = 169.
(169)^1/2 = 13.
13³ = 2197.

(ii) (√[10³ - 6³])³
Answer: 21952
Steps: 1000 - 216 = 784.
√784 = 28.
28³ = 21952.

10. Difference of two perfect cubes is 387. If the cube root of the greater of the two numbers is 8, find the cube root of the smaller number.
Answer: 5
Steps: Let numbers be x and y (x > y).
x - y = 387.
Cube root of x is 8 ⇒ x = 8³ = 512.
512 - y = 387.
y = 512 - 387 = 125.
Cube root of y = √[3]{125} = 5.