Question

What is the smallest non-zero whole number n such that 756n is a perfect cube if 756 equals to 2 square times 3 cube times 7

Answer 1

`\sqrt[3]{756n} = \sqrt[3]{756} \sqrt[3]{n} = \sqrt[3]{27} \sqrt[3]{28} \sqrt[3]{n} = 3 \sqrt[3]{28n}`

`n = {28}^(2) = 784`

The smallest n is 784.

Answer 2

The smallest non-zero whole number n such that 756n is a perfect cube is 49, as multiplying by this number will adjust the power of 7 in the prime factorization of 756 to be a multiple of 3, resulting in a perfect cube.

Step-by-step explanation:

To find the smallest non-zero whole number n such that 756n is a perfect cube, we first need to understand what a perfect cube is.

A perfect cube is a number that can be expressed as the cube of an integer.

The prime factorization of 756 is given as 22 × 33 × 7.

To make 756n a perfect cube, each prime factor must have a power that is a multiple of 3

since a cube has three dimensions and cubing a number multiplies its prime factors three times.

Since 2 is squared and 3 is cubed in the prime factorization, these are already in the right form.

However, 7 is only to the first power.

To make it a perfect cube, we need to multiply by 72 (or 49) as this will give us 7 to the third power.

Therefore, the smallest non-zero whole number n needed is 49.