<s>[INST] Rationalise the denominator and simplify $\frac {3\sqrt 2-4}{3\sqrt2+4}$ Does someone have an idea how to work $\dfrac {3 \sqrt 2 - 4} {3 \sqrt 2 + 4}$ by rationalising the denominator method and simplifying?
 [/INST] For problems like this, you need to multiply numerator and denominator by the denominator's conjugate. A conjugate is found by flipping the sign from positive to negative or negative to positive in a binomial. So, the denominator's conjugate in the above's equation is $ 3\sqrt{2} -4 $. Multiplying an irrational binomial by it's conjugate will rationalize it, as seen below:
$$ \frac{3\sqrt{2}-4}{3\sqrt{2}+4}\times \frac{3\sqrt{2}-4}{3\sqrt{2}-4} $$
Multiply out the bottom and top
$$ \frac{9\times2-12\sqrt{2}-12\sqrt{2}+16}{9\times2 -16} $$
$$ = \frac{18-24\sqrt{2}+16}{18 -16} $$
$$ = \frac{34-24\sqrt{2}}{2} $$
$$ = 17-12\sqrt{2}$$
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