PROBLEMS ON CUBE AND CUBE ROOT

Problem 1 :

The cube root of 125 × 64 is

a.   30     b. 20     c. 40     d. 10

Solution :

∛ (125 × 64) = ∛125 × ∛64

= 5 × 4

= 20

So, option (b) is correct.

Problem 2 :

Volume of cube is 343 cm³, its side is

a.   9 cm     b. 6 cm     c. 9 cm     d. 7 cm

Solution :

Volume of cube = 343 cm³

a³ = 343

a = ∛343

a = 7 cm

So, option (d) is correct.

Problem 3 :

The unit digit of cube of 7812 is

a.   8     b. 4     c. 2     d. 6

Solution :

Unit digit of 7812³ = unit digit of 2³

= unit digit of 8

= 8

So, option (a) is correct.

Problem 4 :

The cube root of 729 is

a.   3     b. 7     c. 9     d. 4

Solution :

= ∛729

= ∛ (9 × 9 × 9)

= 9

So, option (c) is correct.

Problem 5 :

∛ 36 × 6 is

a.    6     b. 4     c. 16     d. 2

Solution :

= ∛ (36 × 6)

= ∛216

= 6

So, option (a) is correct.

Problem 6 :

The cube of -1/2 is

a.    1/4     b. 1/8     c. -1/8     d. -8

Solution :

= (-1/2)³

= -1/8

So, option (c) is correct.

Problem 7 :

By what number should 81 be multiplied to make it a perfect cube

a.   9     b. 1     c. 3     d. -3

Solution :

Prime factor of 81 = (3 × 3 × 3) × 3 x 3 x 3

= 3³ × 3 x 3 x 3

to make it as perfect cube, we need two more 3's. So, the least number by which 81 be multiplied to get a perfect cube is 9.

So, option (a) is correct.

Problem 8 :

Volume of cube whose edge is 4 cm

a.   64cm³     b. 27cm³     c. 2cm³     d. 64cm²

Solution :

Volume of cube = a³

= (4 cm)³

 V = 64 cm³

So, option (a) is correct.

Problem 9 :

Cube root of 0.001 is

a.   -0.1     b. -0.01     c. 0.1     d. -0.13

Solution :

= ∛0.001

= ∛ (0.1)³

= 0.1

So, option (c) is correct.

Problem 10 :

Cube root of (-7)³ × (-3)³ is

a.   -21     b. 21     c. -11     d. 10

Solution :

= ∛ (-7)³ × (-3)³

= -7 × -3

= 21

So, option (b) is correct.

Problem 11 :

Cube of -4/7 is

a.   -64/69    b. -64/49     c. -64/343     d. 64/343

Solution :

= (-4/7)³

= -64/343

So, option (c) is correct.

Problem 12 :

Cube root of 0.001728 is

a.   0.12    b. -0.12     c. -10     d. -0.0012

Solution :

= ∛0.001728

= ∛ (0.12)³

= 0.12

So, option (a) is correct.

Problem 13 :

Three numbers are in the ratio 3: 4: 5. If the sum of their cubes is -1728, find the three numbers.

Solution :

Let three numbers are 3x, 4x, 5x.

Sum of cube = -1728

(3x)³ + (4x)³ + (5x)³ = -1728

27x³ + 64x³ + 125x³ = -1728

216x³ = -1728

x³ = -1728/216

x³ = -8

x = ∛-8

x = -2

Numbers are:

3x = 3(-2) = -6

4x = 4(-2) = -8

5x = 5(-2) = -10

So, the three numbers are -6, -8, and -10.

Problem 14 :

Evaluate ∛5 1182/2197

Solution :

= ∛5 1182/2197

Convert mixed fraction into improper fraction

= ∛ (12167/2197)

= 23/13

Problem 15 :

Evaluate ∛ (-91125 × 512)

Solution :

= ∛ (-91125 × 512)

= ∛ (-45)³ × (8)³

= -45 × 8

= -360

Problem 16 :

Shown is a cube with a volume of 8000cm³

Solution :

Volume of cube = 8000 cm³

a³ = 8000

a = ∛8000

a = ∛(2 x 10 x 2 x 10 x 2 x 10)

= 2 x 10

a = 20 cm

So, the side length of the cube is 20 cm.

Problem 17 :

The number of two cubes are in the ratio of 343 : 1331, the ratio of their edges is :

a) 7: 10      b) 7 : 11     c) 7 : 12      d) 7 : 14

Solution :

The ratio between number of cubes = 343 : 1331

Let a and b be the side length of cubes.

343 : 1331 = a³ : b³

7 x 7 x 7 : 11 x 11 x 11 = a³ : b³

7³ : 11³ = a³ : b³

So, the side lengths of cubes are 7 cm and 11 cm.

Problem 18 :

A 8 x 6 x 4 cm² metallic cube is melted. Find the minimum volume of the molten metal which should be added to mould it into a cube whose edge is an x , when x is an integer

Solution :

Volume of large cube which has the side length of x

Volume of metalic cube = 8 x 6 x 4

= 192 cm³

The nearest perfect cube which is closer to 192 is 216.

 Difference = 216 - 192

= 24

Amount of material to be added is 24 cm³

Problem 19 :

√(5 + ∛x) = 3

a) ∛64     b)  64     c)  √4096     d) ∛262144

Solution :

√(5 + ∛x) = 3

Squaring on both sides, we get

(5 + ∛x) = 3²

(5 + ∛x) = 9

∛x = 9 - 5

∛x = 4

Raising power 3 on both sides,

x = 4³

x = 64

So, option b is correct.