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 :

Problem 11 :

Divide the number 26244 by the smallest number so that the quotient is a perfect cube, so he smallest number is :

a) 4     b) 6      c) 36      d) 16

Solution :

Expressing 26244 as product of prime factors, we get

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

By grouping it as product of three same terms, we need one more 3 and one more 2 to make it as perfect cube.

required numbers to be included = 2 x 3

= 6

So, 6 is the required number to be multiplied by 26244 to make it as perfect cube.

Problem 12 :

The perfect cube nearest to 2750 is :

a) 2749     b) 2747      c) 2744      d) 2754

Solution :

11³ = 1331

12³ = 1728

13³ = 2197

14³ = 2744

15³ = 375

By observing it, 2744 is the perfect cube which is nearest to 2750.

Problem 13 :

Three numbers are in the ratio 1 : 2 : 3. The sum of their cubes is 98784. Find the numbers.

Solution :

Let x, 2x and 3x be the required numbers

x³ + (2x)³ + (3x)³ = 98784

x³ + 8x³ + 27x³ = 98784

36x³ = 98784

x³ = 98784/36

x³ = 2744

x = ∛2744

x = 14

2x = 2(14) ==> 28

3x = 3(14) ==> 42

Then the required numbers are 14, 28 and 42.

Problem 14 :

Three numbers are in the ratio 2 : 3 : 4. The sum of their cubes is 33957. Find the numbers.

Solution :

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

(2x)³ + (3x)³ + (4x)³ = 33957

8x³ + 27x³ + 64x³ = 33957

99x³ = 33957

x³ = 33957/99

x³ = 343

x = ∛343

x = 7

2x ==> 2(14) ==> 28

3x ==>3(14) ==> 42

4x ==> 4(14) ==> 56

So, the required numbers are 28, 42 and 56.

Problem 15 :

Find the smallest number to be multiplied with 3600 will make the product perfect cube. Further find the cube root of the product.

Solution :

Decomposing 3600,

= ∛(2 x 2 x 2 x 2 x 15 x 15)

By grouping these numbers, we see two more 2's and one 15 is needed.

= ∛(2 x 2 x 2 x 2 x 2 x 2 x 15 x 15 x 15)

Required number = 4 x 15

= 60

So, the required number to be multiplied to make it as perfect cube is 60.