simplify 2 divided by 8 - the square root of three

Question

i need to simplify the equation 2 divided by 8- the square root of three.

Answer 1

You might want to say 2 divided by the difference between 8 and the square root of 3 since otherwise your sentence means 2/8 - √3 which is 0.25 - √3.

However, to simplify 2 / (8 - √3), you want to rationalize the denominator.

If you see a + √b (a and b are rational) in the denominator then you want to multiply both sides of the fraction by a - √b, and vice-versa, since the new denominator is then (a + √b)(a - √b) = a² - b, which is rational. This is called rationalization of the denominator.

Let's look at your problem as an example. You have

2 / (8 - √3)                                 Original expression

= 2(8 + √3) / [(8 - √3)(8 + √3)]   Setting up the rationalization of the denominator

= (16 + 2√3) / [8² - (√3)²]           Difference of squares: (a + b)(a - b) = a² - b².

= (16 + 2√3) / 61                        Simplifying the denominator.

= 16/61 + (2/61)√3                     Distributive property (optional step).