Mathmatics Radical Numbers Problem

Question

Hello,

If,

(√5+1) / (√2-1) = x

then,

(√5-1) / (√2+1) = ?

Answer 1

The way the problem is worded is somewhat odd.  It sounds like you're looking to express the second expression in terms of x, rather than just calculate what the second expression is equal to numerically.  But it's not clear.

 

Anyway, here's what I can see doing:

 

x = (√5+1) / (√2-1) =

[(√5+1) (√2+1)] / [(√2-1)(√2+1)] =

(√10 + √5 + √2 + 1)  / [(√2)² - 1²] = 

(√10 + √5 + √2 + 1) / (2 - 1) =

(√10 + √5 + √2 + 1) / 1 =

√10 + √5 + √2 + 1

 

Now let's call the other expression y for the moment...

y = (√5-1) / (√2+1) =

[(√5-1) (√2-1)] / [(√2+1)(√2-1)] =

(√10 - √5 - √2 + 1) / [(√2)² - 1²] =

(√10 - √5 - √2 + 1) / (2 - 1) =

√10 - √5 - √2 + 1

 

So what's the relationship between x and y?

 

Well, x + y = (√10 + √5 + √2 + 1) + (√10 - √5 - √2 + 1) =

√10 + 1

 

So that means that y = √10 + 1 - x.  So you could say that the second expression is equal to √10 + 1 - x.

 

Or , xy = (√10 + √5 + √2 + 1)  (√10 - √5 - √2 + 1) =

(√10 + 1 + √5 + √2)  (√10 + 1 - (√5 + √2)) =

(√10 + 1)² - (√5 + √2)² = 

10 + 1 + 2√10 - (5 + 2 + 2√10) =

11 + 2√10 - 7 - 2√10 = 4

 

xy=4 means that y = 4 / x, so you can say that the second expression is equal to 4/x.

 

You could also get to the second way of doing it in a simpler way.  Again, let y be the second expression.  Now multiply x times y:

 

[(√5+1) / (√2-1)] * [/ (√2+1)] =

[(√5+1) (√5-1)] / [(√2-1) * (√2+1)] =

(5 - 1)((2 - 1) = 4 / 1 = 4

 

So again , xy = 4, so y = 4/x.