3. a) [tex] 7^{2}- 3^{2}+(- \sqrt{5}) ^{2}=49-9+ \sqrt{5} ^{2} =40+5=45 [/tex]
b) [tex] 0^{100}+(- \sqrt{7}) ^{3}+ \sqrt{243}=0+(- \sqrt{7}) ^{3}+9 \sqrt{3} =- \sqrt{7} ^{3}+9 \sqrt{3} =[/tex][tex]- \sqrt{7 ^{3}}+9 \sqrt{3} =-7 \sqrt{7}+9 \sqrt{3} [/tex]
c) [tex] \sqrt{1.44}-(1.2) ^{2}= \sqrt{ \frac{36}{25} }-( \frac{6}{5}) ^{2} = \frac{6}{5}- \frac{36}{25}= \frac{30-36}{25}= \frac{-6}{25}=-\frac{6}{25} [/tex]
d) [tex] \sqrt{3 ^{2} +4 ^{2} } - \sqrt{5 ^{2} }= \sqrt{9+16}-5= \sqrt{25}-5=5-5=0 [/tex]
4. a) [tex][ \frac{3}{ \sqrt{3} }-(0.4+ \sqrt{3})]*10=[ \sqrt{3}-0.4- \sqrt{3}]*10=[-0.4]*10= [/tex][tex]-[0.4*10]=-4[/tex]
b) [tex][-3 \sqrt{5}-( \frac{1}{2}-2 \sqrt{5})]*2= [-3 \sqrt{5}- \frac{1}{2}-2 \sqrt{5}]*2=[- \sqrt{5}- \frac{1}{2}]*2= [/tex][tex]-2 \sqrt{5}- \frac{1}{2}*2=-2 \sqrt{5}-1 [/tex]
c) [tex] \sqrt{( \sqrt{3}- \sqrt{2}) ^{2} } -(\sqrt{3}- \sqrt{2})=\sqrt{3}- \sqrt{2}-\sqrt{3}+ \sqrt{2}=0[/tex]
5. a) [tex] 2^{-1} + 3^{-1}= \frac{1}{2}+ \frac{1}{3}= \frac{3+2}{6}= \frac{5}{6} [/tex]
b) [tex](- \sqrt{2}) ^{-2}+( \sqrt{3}) ^{-2}=- \sqrt{2} ^{-2}+3 ^{-1}=2 ^{-1}+ \frac{1}{3}= \frac{1}{2}+ \frac{1}{3}= \frac{3+2}{6}= \frac{5}{6} [/tex]
c) [tex] \sqrt{27}*( \sqrt{3}) ^{-3}+1= \sqrt{27}\sqrt{3 ^{-3}}+1= \sqrt{27*3 ^{-3} }+1= \sqrt{ 3^{3} *3 ^{-3} }+1= [/tex][tex] \sqrt{1}+1=1+1=2 [/tex]
d) [tex][(2+3) ^{-1}+ \frac{1}{ \sqrt{5} }]:(1+ \sqrt{5})=[5 ^{-1}+ \frac{1}{ \sqrt{5} }]* \frac{1}{1+ \sqrt{5} }=[ \frac{1}{5}+\frac{1}{ \sqrt{5} }]* \frac{1}{1+ \sqrt{5} }= [/tex][tex] \frac{ \sqrt{5}+5 }{5 \sqrt{5} }* \frac{1}{1+ \sqrt{5} }= \frac{( \sqrt{5}+5)*1 }{5 \sqrt{5}*(1+ \sqrt{5}) }= \frac{( \sqrt{5}+5) }{5 \sqrt{5}*(1+ \sqrt{5}) }= \frac{( \sqrt{5}+5) \sqrt{5} }{5 \sqrt{5}*(1+ \sqrt{5}) \sqrt{5} }= [/tex][tex]\frac{( \sqrt{5}+5) \sqrt{5} }{5 *5(1+ \sqrt{5}) }=\frac{( \sqrt{5}+5) \sqrt{5} }{25(1+ \sqrt{5}) }=\frac{5+5 \sqrt{5} }{25(1+ \sqrt{5}) }= \frac{5(1+ \sqrt{5}) }{{25(1+ \sqrt{5}) }}= \frac{1}{5} [/tex]
6. a) [tex]4-4(2 \sqrt{2}- \frac{4}{ \sqrt{2} })=4-4(2 \sqrt{2}- \frac{4 \sqrt{2} }{ 2})=4-4(2 \sqrt{2}-2 \sqrt{2})= [/tex][tex]4-4*0=4-0=4[/tex]
b) [tex][- \sqrt{75}-4( \sqrt{3}-2 \sqrt{3})]:1 ^{50}=[-5 \sqrt{3}-4*(- \sqrt{3})]:1 = [/tex][tex]-5 \sqrt{3}+4 \sqrt{3}=(-5+4) \sqrt3}=-1 \sqrt{3}=- \sqrt{3} [/tex]
c) [tex]5-5*[( \sqrt{6}+ \sqrt{3}): \sqrt{3}- \sqrt{2}]=5-5[ \frac{ \sqrt{6}+ \sqrt{3} }{ \sqrt{3} }- \sqrt{2} ]=[/tex][tex]5- \frac{5( \sqrt{6}+ \sqrt{3}) }{ \sqrt{3} } +5 \sqrt{2}=5- \frac{5 \sqrt{6}+5 \sqrt{3} }{ \sqrt{3} } +5 \sqrt{2}=5- \frac{(5 \sqrt{6}+5 \sqrt{3}) \sqrt{3} }{3}+ 5\sqrt{2}= [/tex][tex]5- \frac{5 \sqrt{18} +15 }{3}+5 \sqrt{2}=5- \frac{15 \sqrt{2} +15 }{3}+5 \sqrt{2}=5- \frac{3(5 \sqrt{2} +5) }{3}+5 \sqrt{2}= [/tex][tex]5-(5 \sqrt{2}+5)+5 \sqrt{2}=5-5 \sqrt{2}-5+5 \sqrt{2}=0 [/tex]
