Question-96460

Question Number 96460 by bobhans last updated on 01/Jun/20

Commented by MJS last updated on 01/Jun/20

−4 only step needed several times is (a/(b+(√c)))=((a(b−(√c)))/(b^2 −c))

$$−\mathrm{4} \\ $$$$\mathrm{only}\:\mathrm{step}\:\mathrm{needed}\:\mathrm{several}\:\mathrm{times}\:\mathrm{is} \\ $$$$\frac{{a}}{{b}+\sqrt{{c}}}=\frac{{a}\left({b}−\sqrt{{c}}\right)}{{b}^{\mathrm{2}} −{c}} \\ $$

Commented by bobhans last updated on 01/Jun/20

$$\mathrm{how}\:\mathrm{to}\:\mathrm{get}\:\mathrm{the}\:\mathrm{short}\:\mathrm{cut}? \\ $$

Commented by MJS last updated on 01/Jun/20

((−4+(√(11)))/(1+(√(−3+(√(11))))))=(((−4+(√(11)))(1−(√(−3+(√(11))))))/(1−(−3+(√(11)))))= =(((−4+(√(11)))(1−(√(−3+(√(11))))))/(4−(√(11))))= =(((−4+(√(11)))(1−(√(−3+(√(11)))))(4+(√(11))))/(16−11))= =(((−16+11)(1−(√(−3+(√(11))))))/(−5))= =1−(√(−3+(√(11)))) the same with the 2nd fraction

$$\frac{−\mathrm{4}+\sqrt{\mathrm{11}}}{\mathrm{1}+\sqrt{−\mathrm{3}+\sqrt{\mathrm{11}}}}=\frac{\left(−\mathrm{4}+\sqrt{\mathrm{11}}\right)\left(\mathrm{1}−\sqrt{−\mathrm{3}+\sqrt{\mathrm{11}}}\right)}{\mathrm{1}−\left(−\mathrm{3}+\sqrt{\mathrm{11}}\right)}= \\ $$$$=\frac{\left(−\mathrm{4}+\sqrt{\mathrm{11}}\right)\left(\mathrm{1}−\sqrt{−\mathrm{3}+\sqrt{\mathrm{11}}}\right)}{\mathrm{4}−\sqrt{\mathrm{11}}}= \\ $$$$=\frac{\left(−\mathrm{4}+\sqrt{\mathrm{11}}\right)\left(\mathrm{1}−\sqrt{−\mathrm{3}+\sqrt{\mathrm{11}}}\right)\left(\mathrm{4}+\sqrt{\mathrm{11}}\right)}{\mathrm{16}−\mathrm{11}}= \\ $$$$=\frac{\left(−\mathrm{16}+\mathrm{11}\right)\left(\mathrm{1}−\sqrt{−\mathrm{3}+\sqrt{\mathrm{11}}}\right)}{−\mathrm{5}}= \\ $$$$=\mathrm{1}−\sqrt{−\mathrm{3}+\sqrt{\mathrm{11}}} \\ $$$$\mathrm{the}\:\mathrm{same}\:\mathrm{with}\:\mathrm{the}\:\mathrm{2nd}\:\mathrm{fraction} \\ $$

Commented by john santu last updated on 01/Jun/20

$$\mathrm{waw}…..\mathrm{great}…. \\ $$

Commented by john santu last updated on 01/Jun/20

second fraction ((−12+(√(11)))/(−3+(√((√(11))−3)))) ×((−3−(√((√(11))−3)))/(−3−(√((√(11))−3)))) = (((−12+(√(11)))(−3−(√((√(11))−3))))/(9−((√(11))−3)))= ((−(12−(√(11)))(−3−(√((√(11))−3))))/(12−(√(11))))= 3+(√((√(11))−3))

$$\mathrm{second}\:\mathrm{fraction}\: \\ $$$$\frac{−\mathrm{12}+\sqrt{\mathrm{11}}}{−\mathrm{3}+\sqrt{\sqrt{\mathrm{11}}−\mathrm{3}}}\:×\frac{−\mathrm{3}−\sqrt{\sqrt{\mathrm{11}}−\mathrm{3}}}{−\mathrm{3}−\sqrt{\sqrt{\mathrm{11}}−\mathrm{3}}}\:= \\ $$$$\frac{\left(−\mathrm{12}+\sqrt{\mathrm{11}}\right)\left(−\mathrm{3}−\sqrt{\sqrt{\mathrm{11}}−\mathrm{3}}\right)}{\mathrm{9}−\left(\sqrt{\mathrm{11}}−\mathrm{3}\right)}= \\ $$$$\frac{−\left(\mathrm{12}−\sqrt{\mathrm{11}}\right)\left(−\mathrm{3}−\sqrt{\sqrt{\mathrm{11}}−\mathrm{3}}\right)}{\mathrm{12}−\sqrt{\mathrm{11}}}= \\ $$$$\mathrm{3}+\sqrt{\sqrt{\mathrm{11}}−\mathrm{3}}\: \\ $$

Commented by john santu last updated on 01/Jun/20

but wrong sir in part (((−4+(√(11)))(1−(√(−3+(√(11))))) )/(4−(√(11)))) = ((−(4−(√(11)))(1−(√(−3+(√(11))))))/(4−(√(11)))) = −1+(√(−3+(√(11))))

$$\mathrm{but}\:\mathrm{wrong}\:\mathrm{sir}\:\mathrm{in}\:\mathrm{part}\: \\ $$$$\frac{\left(−\mathrm{4}+\sqrt{\mathrm{11}}\right)\left(\mathrm{1}−\sqrt{\left.−\mathrm{3}+\sqrt{\mathrm{11}}\right)}\:\right.}{\mathrm{4}−\sqrt{\mathrm{11}}}\:= \\ $$$$\frac{−\left(\mathrm{4}−\sqrt{\mathrm{11}}\right)\left(\mathrm{1}−\sqrt{−\mathrm{3}+\sqrt{\mathrm{11}}}\right)}{\mathrm{4}−\sqrt{\mathrm{11}}}\:= \\ $$$$−\mathrm{1}+\sqrt{−\mathrm{3}+\sqrt{\mathrm{11}}}\: \\ $$

Commented by MJS last updated on 01/Jun/20