Question

Efectuati calculele rationalizand numitorii:

a) $\sqrt{ \frac{288}{27} } + \sqrt{1 \frac{24}{48} } + \sqrt{ 3\frac{27}{72} } - \sqrt{ \frac{1250}{108} }$

b) $\sqrt{ \frac{32}{81} } - \sqrt{12+ \frac{2}{4} } + \sqrt{2+ \frac{26}{36} } - \sqrt{1+ \frac{2}{16} }$

c) $\sqrt{18 \frac{3}{8} }- \sqrt{42+ \frac{2}{3} }+ \sqrt{13 \frac{1}{2} }-( \sqrt{4 \frac{1}{6} }- \sqrt{10 \frac{2}{3} } )$

d) $\frac{4-3 \sqrt{6} }{2 \sqrt{3} } - \frac{5 \sqrt{6}-7 }{4 \sqrt{2} }+ \frac{3+2 \sqrt{6} }{5 \sqrt{3} } - \frac{4 \sqrt{6}-3 }{6 \sqrt{2} }$

Answer 1

a)
√288/27+√1 24/48 +√3 27/72 -√1250/108 =
=√288/27+√72/48 +√243/72 -√1250/108 =
=√32/3+√3/2 +√27/8 -√625/54 =
=4√2/√3+√3/√2 +√27/2√2 -25/3√6 =
=4√6/3+√6/2 +3√6/4 -25√6/18 =
=(48√6+18√6 +27√6 -50√6)/36 =
=43√6/36 

b)
√32/81-√(12+2/4)+√(2+26/36) -√(1+2/16) =
=√32/81-√50/4+√98/36 -√18/16 =
=√32/√81-√50/√4+√98/√36 -√18/√16 =
=4√2/9-5√2/2+7√2/6 -3√2/4 =
=(16√2-90√2+42√2 -27√2)/36 =
=(16√2-90√2+42√2 -27√2)/36 =
=-59√2/36

c)
√18 3/8-√(42+2/3)+√(13 1/2) -(√4 1/6- √10 2/3) =
=√147/8-√128/3+√27/2 -√25/6+ √32/3 =
=7√3/2√2-8√2/√3+3√3/√2 -5/√6+ 4√2/√3 =
=7√6/4-8√6/3+3√6/2 -5√6/6+ 4√6/3 =
=(21√6-32√6+18√6 -10√6+ 16√6)/12 =
=13√6/12

d) (4-3√6)/2√3 -(5√6-7)/4√2 +(3+2√6)/5√3 -(4√6-3)/6√2=
= (4-3√6)√3/6 -(5√6-7)√2/8 +(3+2√6)√3/15 -(4√6-3)√2/12=
= (4√3-3√18)/6 -(5√12-7√2)/8 +(3√3+2√18)/15 -(4√12-3√2)/12=
= (4√3-9√2)/6 -(10√3-7√2)/8 +(3√3+6√2)/15 -(8√3-3√2)/12=
= (4√3-9√2)/6 -(10√3-7√2)/8 +(√3+2√2)/5 -(8√3-3√2)/12=
= (80√3-180√2 -150√3+105√2 +24√3+48√2 -80√3+30√2)/120=
= ( 3√2 -126√3)/120=(√2 -42√3)/40