Hello, 4 little piggies mom!
There's a typo in the first one . . . I'll make a guess.
\(\displaystyle 34)\;\;\sqrt{3}\cdot\left(5\sqrt{3}\,+\,\sqrt{3}\right)\)
Add: \(\displaystyle \,\sqrt{3}\cdot(6\sqrt{3})\)
Multiply: \(\displaystyle \,6\cdot\sqrt{3}\cdot\sqrt{3}\:=\:6\cdot3\:=\:18\)
\(\displaystyle 44)\;\;\frac{6}{10\,+\,\sqrt{2}}\)
Rationalize: \(\displaystyle \L\,\frac{6}{10\,+\,\sqrt{2}}\,\cdot\,\frac{10\,-\,\sqrt{2}}{10\,-\,\sqrt{2}} \;= \;\frac{6(10\,-\,\sqrt{2}}{10^2\,-\,(\sqrt{2})^2}\;=\;\frac{6(10\,-\sqrt{2})}{98} \;= \;\frac{3(10\,-\,\sqrt{2})}{49}\)
\(\displaystyle \L\frac{14}{60\,-\,\sqrt{578}}\)
That radical looks suspiciously reducible . . .
Sure enough: \(\displaystyle \,\sqrt{578}\:=\:\sqrt{289\cdot2}\:=\:\sqrt{289}\cdot\sqrt{2}\:=\:17\sqrt{2}\)
So we have: \(\displaystyle \L\,\frac{14}{60\,-\,17\sqrt{2}}\)
Rationalize: \(\displaystyle \L\,\frac{14}{60\,-\,17\sqrt{2}}\,\cdot\,\frac{60\,+\,17\sqrt{2}}{60\,+\,17\sqrt{2}} \;= \;\frac{14(60\,+\,17\sqrt{2})}{60^2\,-\,(17\sqrt{2})^2} \;= \;\frac{14(60\,+\,17\sqrt{2})}{3600\,-\,578}\)
\(\displaystyle \L\;\;= \;\frac{14(60\,+\,17\sqrt{2})}{3022} \;= \;\frac{7(60\,+\,17\sqrt{2})}{1511}\)
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Some advice:
Do
not multiply out the numerator . . . leave it in factored form.
Get used to the "rationalizing step": \(\displaystyle \,(a\,-\,b)(a\,+\,b)\:=\:a^2\,-\,b^2\)
So that: \(\displaystyle \,(3\,+\,\sqrt{2})(3\,-\,\sqrt{2})\,\) goes
directly to: \(\displaystyle \,3^2\,-\,(\sqrt{2})^2\:=\:9\,-\,2\:=\:7\)
\(\displaystyle \;\;\)
without going through the FOIL each time.
