Question

find the smallest number by which 5184 should be multiplied so that the product is a perfect cube also find the cube root of the product​

Answer 1

Answer:

Given number is 5184

On prime factorising of 5184, we get

5184=2×2×2×2×2×2×3×3×3×3

After grouping the same kind of prime factors is 3’s, its seen that one factor 3 is left ungroup.

5184=(2×2×2)×(2×2×2)×(3×3×3)×3

So, in order to complete it in 3’s, we must multiply the factors 3×3 i.e. equal to 9.

Thus, the required smallest number that should be multiplied to 5184 so that product is a perfect cube is 9.

And,

The cube root of 5184×9=46656 is

=2×2×3×3=36.

Answer 2

Answer:

Find the prime factors:

5184  = 2⁶ x 3⁴

How to find k:

5184  = 2⁶ x 3⁴

For it to be a perfect cube, the exponents should be in the multiple of 3s.

5184k  = 2⁶ x 3⁶

Therefore k = 3² = 9

Find the number:

The number = 5184 x 9 = 46656

Cube root:

46656  = 2⁶ x 3⁶

Divide the exponent by 3 to get the cube root:

∛46656 = 2² x 3² = 4 x 9 = 36

Therefore, The smallest number to be multiplied is 9 and the cube root of the perfect cube is 36.