How do i rationalize (19 - 9*19^(1/2)) / (19+9*19^(1/2))?

Question

(19 - 9*19^(1/2)) / (19+9*19^(1/2))

Answer 1

To rationalize a denominator like (a + b), multiply by (a - b) on top and bottom:

In your case you have:
19 - 9*√19
------------------
19 + 9*√19

So multiply by 19 - 9*√19 on top and bottom:

(19 - 9*√19)²
----------------------------------
(19 + 9*√19)(19 - 9*√19)

The denominator becomes a difference of squares
Remember: (a + b)(a - b) = a² - b²

(19 - 9*√19)²
--------------------
19² + (9*√19)²

Simplify the denominator:
(19 - 9*√19)²
-------------------
1900

Expand out the numerator using FOIL:
(19² - 2(9*√19) + (9 * √19)²)
---------------------------------------
1900

Simplify the numerator:
361 - 18√19 + 9²*19
------------------------------
1900

1900 - 18√19
--------------------
1900

You can factor out a common 2 from top and bottom and cancel it:
950 - 9√19
----------------
950

That's about it...

Answer 2

= (19-9√19)/(19+9√19) * 1
= (19-9√19)/(19+9√19) * (19-9√19)/(19-9√19)
use FOIL method on top and bottom
= (19^2-171√19-171√19+9^2*19) / (19^2-171√19+171√19+9^2*19)
simplify the top and bottom
= (361-342√19+1539) / (361+1539)
= (1900-342√19) / 1900
= 1900/1900 - (342√19)/1900
= 1 - (18√19)/100
= 1 - (9√19)/50

Answer 3

Correctly.

Answer 4

By reading the book and working it out step by step!