Question

a number is 3 more than its reciprocal . what is the difference between their respective cubes

Answer 1

Answer:

The difference between their respective cubes is 36

Step-by-step explanation:

Let the number be x

The reciprocal of the number will be 1/x

According to the question

$x=\frac{1}{x}+3$

$\implies x-\frac{1}{x}=3$

We know that

$a^3-b^3=(a-b)(a^2+ab+b^2)$

or, $a^3-b^3=(a-b)[(a-b)^2+3ab]$

Therefore,

$x^3-(\frac{1}{x})^3=(x-\frac{1}{x})[(x-\frac{1}{x})^2+3x\times\frac{1}{x}]$

$\implies x^3-(\frac{1}{x})^3=3\times[3^2+3]$

$\implies x^3-\frac{1}{x^3}=36$

The difference between the cube of the numbers is 36