Question

How can you write the expression with rationalized denominator? sqrt3 - sqrt6 / sqrt 3 + sqrt 6

Answer 1

Answer:  2√2 - 3

Explanation:


The expession written properly is:

$\frac{ \sqrt{3}- \sqrt{6} }{ \sqrt{3}+ \sqrt{6} }$

To rationalize that kind of expressions, this is to eliminate the radicals on the denominator you use conjugate rationalization.

That is, you have to multiply both numerator and denominator times the conjugate of the denominator.

The conjugate of √3+√6 is √3 - √6, so let's do it:

$\frac{ \sqrt{3} - \sqrt{6} }{ \sqrt{3} + \sqrt{6} } . \frac{ \sqrt{3}- \sqrt{6} }{ \sqrt{3}- \sqrt{6} }$

To help you with the solution of that expression, I will show each part.

1) Numerator: (√3 - √6) . (√3 - √6) = (√3 - √6)^2 = (√3)^2 - 2√3√6 + (√6)^2 =

= 3 - 2√18 + 6 = 9 - 6√2.

2) Denominator: (√3 + √6).(√3 - √6) = (√3)^2 - (√6)^2 = 3 - 6 = - 3

3) Then the resulting expression is:

 9 - 6√2
-----------
 -3

Which can be further simplified, dividing by - 3

    -3 + 2√2

Answer: 2√2 - 3

Answer 2

sqrt 3 - sqrt 6 / sqrt 3 + sqrt 6
can be written as 

sqrt 3 + sqrt 6 - (sqrt 6 / sqrt 3)

I just did some rearranging, no changes yet

Anything times 1 is itself, but the fraction 8/8 is also = 1, and sqrt 67 / sqrt 67 is also 1, so we can multiply anything by anything as long as the number is divided by itself

so we can say


sqrt 3 + sqrt 6 - (sqrt 6 / sqrt 3) = 
sqrt 3 + sqrt 6 - (sqrt 6 / sqrt 3) * (sqrt 3 / sqrt 3)

Now if we write it vertically so its easier to see it becomes


                              (sqrt 6)       (sqrt 3)
 sqrt 3 + sqrt 6  -   ---------    *  ----------  = the original equation
                              (sqrt 3)       (sqrt 3)

on the denominator, sqrt 3 * sqrt 3 AKA (sqrt 3) squared = 3 because the square and the square root cancel out

but the numerator is (sqrt 6) * (sqrt 3) and that's equal to sqrt 18

so we can take a perfect square out of that
18 = 9 * 2 and 9 is a perfect square because
3 * 3 = 9

so we can say that
sqrt 18 = sqrt 9 * sqrt 2 
and sqrt 9 = 3 so
sqrt 18 = 3 * sqrt 2

which is cool because our denominator was sqrt 3 * sqrt 3 which is equal to 3 which now cancels out! So now we have 

                             (sqrt 6)       (sqrt 3)
 sqrt 3 + sqrt 6  -   ---------    *  ----------  = 
                              (sqrt 3)       (sqrt 3)

                             3 (sqrt 2) 
 sqrt 3 + sqrt 6  -   ---------   
                                 3     
3's cancel out and we get

sqrt 3 + sqrt 6 - sqrt 2