how do I rationalize the numerator for:?

Question

[2sqrt(5) - 3sqrt(2)] / [2sqrt(5) + 3sqrt(2)]

[2sqrt(5) - 3sqrt(2)] / [2sqrt(5) + 3sqrt(2)]

I have o rationalize the numerator not the denominator.

Answer 1

Problems like this arise in calculus, for example
when one calculates the derivative of sqrt(x).
To do it, multiply top and bottom by 2sqrt(5) + 3sqrt(2).
You get 2/ (38 + 12sqrt(10) ).
Knowing how to rationalise both numerators
and denominators will be important in your later work.

Answer 2

I agree with the others, that normally you rationalize the denominator. But if you want to do the numerator...

Remember that (x + y)(x - y) = x^2 - y^2

So we are going to start with something like (a * sqrt(5) + b * sqrt(2))

Now we also know that (a * -3) + (b * 2) = 0.
We can make that happen by setting a to 2 and b to 3 so that we will get -6 + 6 = 0

So (2sqrt(5) + 3sqrt(2)) * (2sqrt(5) - 3sqrt(2))

If we solve this, we will get
(2sqrt(5) * 2sqrt(5)) + (2sqrt(5) * -3sqrt(2)) + (3sqrt(2) * 2sqrt(5)) + (3sqrt(2) * 3sqrt(2))

4sqrt(5)sqrt(5) - 6sqrt(5)sqrt(2) + 6sqrt(5)sqrt(2) + 9sqrt(2)sqrt(2)

4 * 5 + 9 * 2

20 + 18

38

Answer 3

You must be going to publik skool. There is no point to rationalize the numerator, but you do it by multiplying the top and bottom of the fraction by 2sqrt(5) + 3sqrt(2). The resulting numerator is 38. The denominator gets to be 38 + 6sqrt(10).

Answer 4

you don't rationalize the numerator. The denom would be multiplied by its conjugate or 2sqrt(5)- 3 sqrt(2)
multiply top and bottom and get

[56-6*sqrt(10)] /(-16)

Answer 5

(2sqrt(5) - 3sqrt(2))/(2sqrt(5) + 3sqrt(2))

(sqrt(20) - sqrt(18))/(sqrt(20) + sqrt(18))

Multiply top and bottom by sqrt(20) - sqrt(18)

((sqrt(20) - sqrt(18))(sqrt(20) - sqrt(18))/((sqrt(20) + sqrt(18))(sqrt(20) - sqrt(18))

(20 - sqrt(360) - sqrt(360) + 18)/(20 - sqrt(360) + sqrt(360) - 18)

(38 - 2sqrt(360))/2

(38 - 2sqrt(36 * 10))/2

(38 - 12sqrt(10))/2

ANS : 19 - 6sqrt(10)