SIMPLIFICATION PROBLEMS ON CUBE ROOTS

Problem 1 :

Find the value of following cube roots:

x = ∛(27 × 2744)

Solution :

x = ∛(27 × 2744)

x = ∛(3 x 3 x 3 × 2 x 2 x 2 x 7 x 7 x 7)

x = 3 x 2 x 7

x = 42

Problem 2 :

Evaluate 

Solution :

Problem 3 :

Two numbers 4x and 5x are such that sum of their cubes is 189. Find x.

Solution :

Sum of their cubes = 189

(4x)³ + (5x)³ = 189

64x³ + 125x³ = 189

189 x³ = 189

x³ = 189/189

x³ = 1

x = 1

Problem 4 :

The volume of a cube is 5832 m³, find the length of the side.

Solution :

Volume of cube = 5832 m³

a³ = 5832

a = ∛5832

a = ∛(2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 3)

a = 2 x 3 x 3

a = 18

So, side length of cube is 18 m.

Problem 5 :

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

Solution :

Let the three sides be 3x, 4x and 5x.

Sum of their cubes = -1728

(3x)³ + (4x)³ + (5x)³ = -1728

(27+64+125)x³ = -1728

216x³ = -1728

x³ = -1728/216

x³ = -8

x = -2

Problem 6 :

Check whether 648 is a perfect cube or not.

Solution :

Decomposing 648,

648 = 2 x 2 x 2 x 3 x 3 x 3 x 3

We see three 2's, three 3's. There is one more 3 not grouped.

So, the given number is not a perfect cube.

Problem 7 :

Three numbers are to one another as 2:3:4. The sum of their cubes is 33957. Find the numbers.

Solution : 

Let the three numbers be 2x, 3x and 4x

(2x)³ + (3x)³ + (4x)³ = 33957

(8 + 27 + 64)x³ = 33957

99x³ = 33957

x³ = 33957/99

x³ = 343

x = 7

2x ==> 2(7) ==> 14

3x ==> 3(7) ==> 21

4x ==> 4(7) ==> 28

So, the three numbers are 14, 21 and 28.

Problem 8 :

Solution :

Problem 9 :

If y = 5, then what is the value of 10y√(y³ - y²) ?

Solution :

= 10y√(y³ - y²)

By applying the value of y, we get

= 10(5)√(5³ - 5²)

= 50√(125 - 25)

= 50√100

= 50 (10)

= 500

Problem 10 :

Solution :