express with rational denominator

1 / 1 + √3 + √5 + √15

and expand:

(√x -1 + 1/√x)^2

Originally Posted by richardkim4668
express with rational denominator

1 / 1 + √3 + √5 + √15

and expand:

(√x -1 + 1/√x)^2
need to ask you to wrap things in parentheses to make things clear. do you mean

$\dfrac 1 {1+\sqrt{3}+\sqrt{5}+\sqrt{15}}$ ?

for the second one

$(\sqrt{x}-1+\dfrac 1 {\sqrt{x}})^2 =$

$(\sqrt{x})^2-\sqrt{x}+\sqrt{x}\dfrac 1 {\sqrt{x}} - \sqrt{x} + 1 - \dfrac 1 {\sqrt{x}} + \dfrac 1 {\sqrt{x}} \sqrt{x} - \dfrac 1 {\sqrt{x}} + \left(\dfrac 1 {\sqrt{x}}\right)^2=$

$x - 2\sqrt{x} + 3 - \dfrac 2 {\sqrt{x}} + \dfrac 1 x=$

$\dfrac {x^2 - 2x^{3/2} + 3x - 2\sqrt{x} + 1} x$

yup i mean that

Originally Posted by romsek
need to ask you to wrap things in parentheses to make things clear. do you mean

$\dfrac 1 {1+\sqrt{3}+\sqrt{5}+\sqrt{15}}$ ?
Originally Posted by richardkim4668
yup i mean that
$\dfrac 1 {1+\sqrt{3}+\sqrt{5}+\sqrt{15}}=$

$\dfrac 1 {1+\sqrt 3 + \sqrt 5(1 + \sqrt 3)} =$

$\dfrac 1 {(\sqrt 3 + 1)(\sqrt 5 + 1)} =$

$\dfrac 1 {(\sqrt 3 + 1)(\sqrt 5 + 1)}\dfrac{\sqrt 5 - 1}{\sqrt 5 - 1}=$

$\dfrac{\sqrt 5 - 1}{(\sqrt 3 +1)(4)}=$

$\dfrac{\sqrt 5 - 1}{(\sqrt 3 +1)(4)}\dfrac{\sqrt 3 -1}{\sqrt 3 - 1}=$

$\dfrac{(\sqrt 5 - 1)(\sqrt 3 - 1)}{(2)(4)}=$

$\dfrac{(\sqrt 5 - 1)(\sqrt 3 - 1)} 8$

$\dfrac {\sqrt {15} - \sqrt 3 - \sqrt 5 + 1} 8$