Question

Rationalize the denominator and simplify.

a) 5/4- √3
b) 12/√7 + 2
c) 6/√2 - √3
d) 15√6/3√5

Answer 1

lemme do the first one
what you do is just grab the denominator's CONJUGATE, that is ,the same expression but with a different sign in between

multiply top and bottom by it, and then \(\bf \cfrac{5}{4-\sqrt{3}}\cdot \cfrac{{\color{blue}{ 4+\sqrt{3}}}}{{\color{blue}{ 4+\sqrt{3}}}}\implies \cfrac{5(4+\sqrt{3})}{(4-\sqrt{3})(4+\sqrt{3})}
\\ \quad \\
\textit{keep in mind that }{\color{blue}{ (a-b)(a+b) = a^2-b^2}}\qquad thus
\\ \quad \\
\cfrac{5(4+\sqrt{3})}{(4-\sqrt{3})(4+\sqrt{3})}\implies \cfrac{5(4+\sqrt{3})}{4^2-(\sqrt{3})^2}\implies \cfrac{5(4+\sqrt{3})}{16-3}\)

Answer 2

when you do not have a binomial, like in D) you simply multiply top and bottom by the denominator itself, that way it squares up and the radicand comes out