Section 6.4
Radical Simplification 2
The Distributive Axiom is important when simplifying the sum of several radicals. It is important to recognize the difference between $3\sqrt{5}+7\sqrt{5}$, which can be simplified, since it is the same as $\left( 3+7 \right)\sqrt{5}$ (or $10\sqrt{5}$), and $\sqrt{5}+\sqrt{7}$, which cannot be simplified, since $\sqrt{5}$ and $\sqrt{7}$ are not alike.
Examples:
a. $\sqrt{8}+\sqrt{18}-\sqrt{50}=2\sqrt{2}+3\sqrt{2}-5\sqrt{2}=0$
b. $2+3\sqrt{5}$ cannot be simplified. This is not the same as $\left( 2+3 \right)\sqrt{5}$. The Order of Operations rules.
$\begin{align}
& \text{c. }\left( \sqrt{3}-\sqrt{2} \right)\left( \sqrt{6}+2\sqrt{2} \right)=\sqrt{18}-\sqrt{12}+2\sqrt{6}-2\sqrt{4} \\
& =3\sqrt{2}-2\sqrt{3}+2\sqrt{6}-4
\end{align}$
Expressions of the form $a+b$ and $a-b$ are known as "conjugates" of each other. Thus $2+\sqrt{3}$ and $2-\sqrt{3}$ are conjugates of each other. Conjugates with square roots have the nice property of multiplying together to produce products that have no square roots. This is particularly useful in simplifying a fraction with a denominator such as $2-\sqrt{3}$.
d. $\left( 2-\sqrt{3} \right)\left( 2+\sqrt{3} \right)=4-3=1$
e. $\left( \sqrt{6}-\sqrt{2} \right)\left( \sqrt{6}+\sqrt{2} \right)=6-2=4$
f. $\frac{14}{3+\sqrt{5}}=\frac{14\left( 3-\sqrt{5} \right)}{\left( 3+\sqrt{5} \right)\left( 3-\sqrt{5} \right)}=\frac{14\left( 3-\sqrt{5} \right)}{9-5}=\frac{14\left( 3-\sqrt{5} \right)}{4}=\frac{7\left( 3-\sqrt{5} \right)}{2}$
Adding and simplifying radicals: More Simplifying Radical Expressions
Exercises:
Simplify each of the following. All letters represent positive numbers.
\begin{array}{l l l}
1. \;\;\;\sqrt{8}-\sqrt{50}+\sqrt{72}\;\;\; & 2. \;\;\;\sqrt{27}-\sqrt{12}+\sqrt{75}\;\;\; & 3. \;\;\;\sqrt{6}\left( \sqrt{18}-\sqrt{12} \right)\;\;\; \\
4. \;\;\;\frac{1}{\sqrt{3}}+\sqrt{3}\;\;\; & 5. \;\;\;\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\;\;\; & 6. \;\;\;\sqrt{6}\cdot \left( \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}} \right)\;\;\; \\
7. \;\;\;\sqrt{8x}+\sqrt{2x}+\sqrt{18x}\;\;\; & 8. \;\;\;\sqrt{12}+\sqrt{75}-9\sqrt{\frac{1}{3}}\;\;\; & 9. \;\;\;\sqrt{\frac{3}{5}}+\sqrt{\frac{5}{3}}+\frac{2}{\sqrt{15}}\;\;\; \\
10. \;\;\;\frac{6}{\sqrt{3}}-\sqrt{\frac{100}{27}}+\frac{1}{3}\sqrt{\frac{1}{3}}\;\;\; & 11. \;\;\;\sqrt{\frac{2}{3}}+\sqrt{\frac{3}{2}}\;\;\; & 12. \;\;\;{{\left( 1+2\sqrt{2} \right)}^{2}}-{{\left( 1-2\sqrt{2} \right)}^{2}}\;\;\; \\
13. \;\;\;\frac{3}{\sqrt{8}}-\frac{2}{\sqrt{18}}\;\;\; & 14. \;\;\;\sqrt{\frac{3}{5}}\div \frac{\sqrt{3}}{5}\;\;\; & 15. \;\;\;\frac{\sqrt{2}}{3}+\frac{3}{\sqrt{2}}\;\;\; \\
16. \;\;\;\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}\;\;\; & 17. \;\;\;\left( 7+\sqrt{2} \right)\left( 7-\sqrt{2} \right)\;\;\; & 18. \;\;\;\left( 5+\sqrt{6} \right)\left( 5-\sqrt{6} \right)\;\;\; \\
19. \;\;\;\left( 2+\sqrt{2} \right)\left( 2-\sqrt{2} \right)\;\;\; & 20. \;\;\;\left( \sqrt{5}+\sqrt{2} \right)\left( \sqrt{5}-\sqrt{2} \right)\;\;\; & 21. \;\;\;\left( 3\sqrt{7}+\sqrt{5} \right)\left( 3\sqrt{7}-\sqrt{5} \right)\;\;\; \\
22. \;\;\;{{\left( 2+\sqrt{3} \right)}^{2}}\;\;\; & 23. \;\;\;{{\left( 4-\sqrt{2} \right)}^{2}}\;\;\; & 24. \;\;\;{{\left( 1-\sqrt{2} \right)}^{2}}\;\;\; \\
25. \;\;\;{{\left( \sqrt{3}-\sqrt{2} \right)}^{2}}\;\;\; & 26. \;\;\;{{\left( \sqrt{6}-\sqrt{5} \right)}^{2}}\;\;\; & 27. \;\;\;2\sqrt{48}-\sqrt{50}+\frac{3}{\sqrt{3}-\sqrt{2}}\;\;\; \\
28. \;\;\;{{\left( 3-\sqrt{5} \right)}^{2}}\;\;\; & 29. \;\;\;\frac{2}{-1+\sqrt{5}}\;\;\; & 30. \;\;\;\frac{2}{1+\sqrt{5}}\;\;\; \\
31. \;\;\;\frac{2}{1+\sqrt{5}}-\frac{2}{1-\sqrt{5}}\;\;\; & 32. \;\;\;\frac{14}{\sqrt{5}+\sqrt{3}}\;\;\; & 33. \;\;\;\frac{5-x}{\sqrt{5}+\sqrt{x}}\;\;\; \\
34. \;\;\;\frac{7-2x}{\sqrt{7}+\sqrt{2x}}\;\;\; & 35. \;\;\;\frac{a-b}{\sqrt{a}+\sqrt{b}}\;\;\; & 36. \;\;\;{{\left( \sqrt{2}+\sqrt{3} \right)}^{2}}\;\;\; \\
37. \;\;\;\frac{3+\sqrt{2}}{3-\sqrt{2}}\;\;\; & 38. \;\;\;\frac{2-\sqrt{2}}{2+\sqrt{2}}\;\;\; & 39. \;\;\;\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}\;\;\; \\
40. \;\;\;\frac{a+\sqrt{b}}{a-\sqrt{b}}\;\;\; & 41. \;\;\;{{\left( \frac{1+\sqrt{5}}{2} \right)}^{2}}\;\;\; & 42. \;\;\;{{\left( \frac{1+\sqrt{3}}{2} \right)}^{2}}\;\;\; \\
\end{array}
\begin{array}{l l}
43. \;\;\;\left( \left( 1+\sqrt{2} \right)+\sqrt{3} \right)\left( \left( 1+\sqrt{2} \right)-\sqrt{3} \right)\;\;\; & 44. \;\;\;\left( \left( \sqrt{3}+\sqrt{2} \right)+\sqrt{5} \right)\left( \left( \sqrt{3}+\sqrt{2} \right)-\sqrt{5} \right)\;\;\; \\
45. \;\;\;\left( \left( 1+\sqrt{5} \right)+\sqrt{6} \right)\left( \left( 1+\sqrt{5} \right)-\sqrt{6} \right)\;\;\; & 46. \;\;\;\left( \left( \sqrt{a}+\sqrt{b} \right)+\sqrt{a+b} \right)\left( \left( \sqrt{a}+\sqrt{b} \right)-\sqrt{a+b} \right)\;\;\; \\
47. \;\;\;\left( \sqrt{2}+\sqrt{18}+\sqrt{12} \right)\left( \sqrt{2}+\sqrt{18}-\sqrt{12} \right)\;\;\; & 48. \;\;\;\left( \left( \sqrt{a}+\sqrt{b} \right)+\sqrt{2\sqrt{ab}} \right)\left( \left( \sqrt{a}+\sqrt{b} \right)-\sqrt{2\sqrt{ab}} \right)\;\;\;
\end{array}