Question

How do you rationalize the denominator and simplify `(12-2sqrt6)/(8-5sqrt2)`?

Answer 1

See the explanation

Explanation:

`color(blue)("Preamble")`

There is a trick to this. Apart from anything else, mathematicians do not like 'roots' in the denominator. It is considered good practice to 'get rid of them'.

The trick: multiply by its self but change the sign in the middle (for the second variable) .

Why? Consider: `(a-b)(a+b) = a^2-b^2`

Observe that everything ends up being squared. So any square roots disappear.
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`color(blue)("Answering the question")`

`color(green)((12-2sqrt6)/(8-5sqrt2)color(red)(xx1)" "=" "color(green)((12-2sqrt6)/(8-5sqrt2)color(red)(xx(8+5sqrt(2))/(8+5sqrt(2)) )`

`=[(12-2sqrt(6))(8+5sqrt(2))]/ [8^2-(5sqrt(2))^2 ]`

`=(96+60sqrt(2)-64sqrt(6)-10sqrt(12))/(64-50 )`
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Consider `10sqrt(12) -> 10sqrt(2^2xx3) = 20sqrt(3)`
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`=(96+60sqrt(2)-64sqrt(6)-20sqrt(3))/(14 )`

`=(48+30sqrt(2)-32sqrt(6)-5sqrt(12))/7`

`color(blue)("I will leave the rest for you to do")`