Question

Rationalize the denominator and simplify.

a) 12/√7 + 2
b) 6/√2 - √3
c) 15√6/3√5

Answer 1

use the conjugate and multiply top and bottom by the conjugate of the denominator, for the first 2
the last one, just multiply by root 5 \(\bf \cfrac{12}{\sqrt{7}+2}\cdot \cfrac{\sqrt{7}-2}{\sqrt{7}-2}\implies \cfrac{12(\sqrt{7}-2)}{(\sqrt{7}+2)(\sqrt{7}-2)}
\\ \quad \\
recall\implies {\color{blue}{ (a-b)(a+b) = a^2-b^2}}\qquad thus
\\ \quad \\
\cfrac{12(\sqrt{7}-2)}{(\sqrt{7}+2)(\sqrt{7}-2)}\implies \cfrac{12(\sqrt{7}-2)}{(\sqrt{7})^2-2^2}\\
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\cfrac{6}{\sqrt{2}-\sqrt{3}}\cdot \cfrac{\sqrt{2}+\sqrt{3}}{\sqrt{2}+\sqrt{3}}\implies \cfrac{6(\sqrt{2}+\sqrt{3})}{(\sqrt{2}-\sqrt{3})(\sqrt{2}+\sqrt{3})}\implies \cfrac{6(\sqrt{2}+\sqrt{3})}{(\sqrt{2})^2-(\sqrt{3})^2}\\
---------------------------\\
\cfrac{15\sqrt{6}}{3\sqrt{5}}\cdot \cfrac{3\sqrt{5}}{3\sqrt{5}}\implies \cfrac{(15\sqrt{6})(3\sqrt{5})}{3(\sqrt{5})^2}\)

expand and simplify