CUBE ROOTS WORKSHEET

1. Find the cube root of 8.

Solution :

³√8 = ³√(2 x 2 x 2)

= 2

2. Find the cube root of 27.

Solution :

³√27 = ³√(3 x 3 x 3)

= 3

3. Find the cube root of 512.

Solution :

³√512 = ³√(2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2)

= ³√[(2 x 2 x 2)(2 x 2 x 2)(2 x 2 x 2)]

= 2 x 2 x 2

= 8

4. Find the cube root of (27 x 64).

Solution :

³√(27 x 64) = ³√[(3 x 3 x 3)(2 x 2 x 2 x 2 x 2 x 2)]

= ³√[(3 x 3 x 3)(2 x 2 x 2)(2 x 2 x 2)]

= ³√[(3 x 3 x 3)(2 x 2 x 2)(2 x 2 x 2)]

= 3 x 2 x 2

= 12

5. Find the cube root of 1000.

Solution :

³√1000 = ³√(2 x 2 x 2 x 5 x 5 x 5)

=³√[(2 x 2 x 2)(5 x 5 x 5)]

= 2 x 5

= 10

Problem 6 :

Find the cube root of (8/125).

Solution :

³√(8/125) = ³√8/³√125

=³√(2 x 2 x 2)/³√(5 x 5 x 5)

= 2/5

Problem 7 :

Find the cube root of 0.008.

Solution :

³√0.008 = ³√(8/1000)

=³√8/³√1000

= 2/10

= 0.2

Problem 8 :

Find the cube root of 0.027.

Solution :

³√0.027 = ³√(27/1000)

=³√27/³√1000

= 3/10

= 0.3

Problem 9 :

Find the cube root of 0.343.

Solution :

³√0.343 = ³√(343/1000)

=³√343/³√1000

= 7/10

= 0.7

Problem 10 :

Find the cube root of -125.

Solution :

³√-125 = ³√(-5 x -5 x -5)

= -5

Problem 11 :

∛(1 +  ∛343) is equal to

a)  2     b)  3    c)  6     d) 7

Solution :

∛(1 +  ∛343) = ∛((1 ⋅ 1 ⋅ 1)  +  ∛(7 ⋅ 7 ⋅ 7))

= ∛(1 + 7)

= ∛8

= ∛(2 ⋅ 2 ⋅ 2)

= 2

Problem 12 :

The smallest number, by which 675 is multiplied so that the product if s perfect cube is

a)  3     b)  5    c)  40     d) 125

Solution :

By decomposing 675, we get

675 = 5 x 5 x 3 x 3 x 3

To make it as group of three same terms, there is one 5 insufficient. So, we need to have one more 5 to make it as perfect cube. So, the required smallest number to be multiplied by 675 is 5.

Problem 13 :

Which of the following numbers is not a perfect cube ?

a)  216      b)  567    c)  125     d) 343

Solution :

216 = 6 x 6 x 6

343 = 7 x 7 x 7

125 = 5 x 5 x 5

Here 567 cannot be written as a product of three same terms. So, option b is not a perfect cube.

Problem 14 :

∛185193 = 57, then the value of 

∛185193 + ∛185.193 + ∛0.000185193

a)  6.327    b) 63.275      c) 632.75     d)  62.757 

∛185193 = 57

∛185.193 = ∛(185193 / 1000)

= 57/10

= 5.7

 ∛0.000185193 = ∛(185193 / 10⁹)

= 57/10³

= 57/1000

= 0.057

∛185193 + ∛185.193 + ∛0.000185193 = 57 + 5.7 + 0.057

= 62.757

Problem 15 :

The value of (∛64 + ∛125)/∛27 is 

a)  2     b)  3    c)  4    d)  9

Solution :

= (∛64 + ∛125)/∛27

∛64 = ∛(4 ⋅ 4 ⋅ 4) = 4

∛125 = ∛(5 ⋅ 5 ⋅ 5) = 5

∛27 = ∛(3 ⋅ 3 ⋅ 3) = 3

= (4 + 5) / 3

= 9/3

= 3

So, option b is correct.

Problem 16 :

Find the length of each side of a cube if its volume is 512 cm³.

Solution :

Volume of cube = 512 cm³

Let x be the side length of cube

x³ = 512

x = ∛512

= ∛(8 ⋅ 8 ⋅ 8)

= 8 cm

So, side length of cube is 8 cm.

Problem 17 :

What is the smallest number by which 1323 may be multiplied so that the product is a perfect cube?

Solution :

Decomposing 1323, we get 

1323 = 3 x 3 x 3 x 7 x 7

By grouping three same values, there is two 7's. To group 7, we need one more 7.

To make it as perfect cube, there should be one 7 multiplied.

Problem 18 :

What is the smallest number by which 1375 should be divided so that the quotient may be a perfect cube ?

Solution :

1375 = 5 x 5 x 5 x 11

By grouping three same values, there is one 11.  To make it as perfect cube, there is one 11 extra. So, 11 should be divided to make it as perfect cube.

Problem 19 :

Sujatha makes a cuboid of plasticine of sides 5 cm, 2 cm, 5 cm. How many minimum such cuboids will she need to form a cube?

Solution :

Volume of cuboid = length x width x height

length = 5 cm, width = 2 cm and height = 5 cm

Volume = 5 x 2 x 5

= 50

Number of cube needed = volume of cuboid / volume of cube

Volume of cube = 50/n

To make 50 as perfect cube, we need to multiply both numerator and denominator by 20.

= 50 x (20/20)

= (1000/20)

so, the required number of cubes is 20.

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