Answer:
[tex]\huge\boxed{\sqrt{15}+1}[/tex]
Step-by-step explanation:
We can simplify the radical expression:
[tex]\dfrac{12\sqrt{5}-38\sqrt{3}}{7\sqrt{3}-9\sqrt{5}}[/tex]
by multiplying the numerator and denominator by the conjugate of the denominator.
The conjugate of a binomial [tex]a-b[/tex] is:
[tex]a+b[/tex]
So, we need to multiply our numerator and denominator by:
[tex]7\sqrt{3}+9\sqrt{5}[/tex]
↓↓↓
[tex]\dfrac{12\sqrt{5}-38\sqrt{3}}{7\sqrt{3}-9\sqrt{5}}\cdot \dfrac{7\sqrt{3}+9\sqrt{5}}{{7\sqrt{3}+9\sqrt{5}}}[/tex]
↓ applying the distributive property ... [tex](b+c)\cdot a = b(a) + c(a)[/tex]
[tex]\dfrac{\left(\dfrac{}{}12\sqrt{5}\dfrac{}{}\right)(7\sqrt{3}+9\sqrt{5})-\left(\dfrac{}{}38\sqrt{3}\dfrac{}{}\right)(7\sqrt{3}+9\sqrt{5})}{\left(\dfrac{}{}7\sqrt{3}\dfrac{}{}\right)(7\sqrt{3}+9\sqrt{5})-\left(\dfrac{}{}9\sqrt{5}\dfrac{}{}\right)(7\sqrt{3}+9\sqrt{5})}[/tex]
↓ multiplying out
[tex]\dfrac{\left(84\sqrt{15} + 108(5)\right)-\left(266(9)+342\sqrt{15}\,\right)}{\left(49(3) + 63\sqrt{15}\, \right)-\left(63\sqrt{15} + 81(5)\right)}[/tex]
↓ distributing the negatives ... [tex]a-(b+c) = a-b - c[/tex]
[tex]\dfrac{84\sqrt{15} + 108(5)-266(3)-342\sqrt{15}}{49(3) + 63\sqrt{15}-63\sqrt{15} - 81(5)}[/tex]
↓ combining like terms
[tex]\dfrac{-258\sqrt{15} - 258}{-258}[/tex]
↓ splitting into partial fractions ... [tex]\dfrac{a+b}{c} = \dfrac{a}{c}+\dfrac{b}{c}[/tex]
[tex]\dfrac{-258\sqrt{15}}{-258} + \dfrac{(-258)}{-258}[/tex]
↓ executing the division
[tex]\boxed{\sqrt{15}+1}[/tex]
