Answer:
Step-by-step explanation:
To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator.
Conjugate of 4√3 + √2 is 4√3 - √2
(2√3 +3 √2)(4√3 - √2) = (2√3 * 4√3) - (2√3*√2) + (3√2*4√3) -(3√2*√2) {Use FOIL method}
= 8*3 - 2*√6+12√6 -3*2
= 24 -2√6 +12√6 - 6
= 24 - 6 + (-2+12)√6)
= 18 + 10√6
[tex]\frac{2\sqrt{3}+3\sqrt{2} }{(4\sqrt{3}+\sqrt{2} }\\\\=\frac{(2\sqrt{3}+3\sqrt{2})(4\sqrt{3}-\sqrt{2}) }{(4\sqrt{3}+\sqrt{2} )(4\sqrt{3}-\sqrt{2} )}\\\\=\frac{18+10\sqrt{6} }{(4\sqrt{3})^{2}-(\sqrt{2})^{2} }\\\\=\frac{18+10\sqrt{6} }{48-2}\\\\=\frac{18+10\sqrt{6} }{46}\\\\\\=\frac{2(9+5\sqrt{6} )}{46}\\\\=\frac{9+5\sqrt{6} }{23}[/tex]
