HOW TO FIND THE CUBE ROOT OF A NUMBER USING PRIME FACTORIZATION

Example 1 :

Find the cube root of 8.

Solution :

³√8 = ³√(2 x 2 x 2)

= 2

Example 2 :

Find the cube root of 27.

Solution :

³√27 = ³√(3 x 3 x 3)

= 3

Example 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

Example 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

Example 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

Example 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

Example 7 :

Find the cube root of 0.008.

Solution :

³√0.008 = ³√(8/1000)

=³√8/³√1000

= 2/10

= 0.2

Example 8 :

Find the cube root of 0.027.

Solution :

³√0.027 = ³√(27/1000)

=³√27/³√1000

= 3/10

= 0.3

Example 9 :

Find the cube root of 0.343.

Solution :

³√0.343 = ³√(343/1000)

=³√343/³√1000

= 7/10

= 0.7

Example 10 :

Find the cube root of -125.

Solution :

³√-125 = ³√(-5 x -5 x -5)

= -5

Example 11 :

Find the cube root of -125.

Solution :

³√-125 = ³√(-5 x -5 x -5)

= -5

Example 12 :

What is the smallest number by which 4608 may be multiplied so that the product is perfect cube?

Solution :

Decomposing 4608, we get

4608 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 2 x 2

By writing the repeated factors in exponential form, we understand that how many groups of three same numerals are there.

= 2⁹ x 3²

= 2³ x 2³ x 2³ x 3²

Since there is one more 3 is needed to make it as three 3's, the required smallest number of be multiplied to make 4068 as perfect cube is 3.

Example 13 :

What is the smallest number by which 2304 may be divided so that the equation is a perfect cube?

Solution :

2304 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3

= 2⁸ x 3²

Grouping into three same values,

= 2⁶ x (2² x 3²)

= 2⁶ x (4 x 9)

= 2⁶ x 36

So, 36 is the number to be divided to make it as perfect cube.

Example 14 :

Find the surface area of a cube whose volume is 343 m³

Solution :

Volume of cube = 343 m³

Let a be the side length of cube.

a³ = 343

a³ = 7 x 7 x 7

a³ = 7³ 

a = 7

So, side length of cube is 7 m

Surface area of cube = 6a²

= 6(7²)

= 6(49)

= 294 m²

Example 15 :

Find the cube root of 3375 × (−729)

Solution :

Cube root of (3375 × (−729))

= ³√(3375 × (−729))

= ³√(5 x 5 x 5 x 3 x 3 x 3) × (−9 x (-9) x (-9))

Taking out one value for every three same values, we get

= 5 x 3 x (-9)

= -135

So, the cube root of the given number is -135.

Example 16 :

Is 1188 a perfect cube? If not, by which smallest natural number should 1188 be divided so that the quotient is a perfect cube? 

Solution :

To check 1188 is a perfect cube or not, we decompose the given number as much as possible.

1188 = 2 x 2 x 3 x 3 x 3 x 11

= 3³ x 2² x 11

= 3³ x (4 x 11)

Here 44 is the number which is extra. So, 1188 should be divided by 44 to make it perfect cube.

Example 17 :

A cubical box has a volume of 512000 cubic cm. What is the length of the side of box?

Solution :

Volume of cubical box = 512000 cubic cm

Let a be the side length of cube.

a³ = 512000

a³ = (8 x 8 x 8 x 10 x 10 x 10)

a³ = 8³ x 10³

= (8 x 10)³

a³ = (80)³

a = 80 cm

So, side length of cube is 80 cm.

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