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

Commented by MJS last updated on 01/Jun/20

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

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

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

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

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

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

$$\mathrm{second}\mathrm{fraction}$$$$\frac{-12+\sqrt{11}}{-3+\sqrt{\sqrt{11}-3}}\times \frac{-3-\sqrt{\sqrt{11}-3}}{-3-\sqrt{\sqrt{11}-3}}=$$$$\frac{(-12+\sqrt{11})(-3-\sqrt{\sqrt{11}-3})}{9-(\sqrt{11}-3)}=$$$$\frac{-(12-\sqrt{11})(-3-\sqrt{\sqrt{11}-3})}{12-\sqrt{11}}=$$$$3+\sqrt{\sqrt{11}-3}$$

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

$$\mathrm{but}\mathrm{wrong}\mathrm{sir}\mathrm{in}\mathrm{part}$$$$\frac{(-4+\sqrt{11})(1-\sqrt{-3+\sqrt{11})}}{4-\sqrt{11}}=$$$$\frac{-(4-\sqrt{11})(1-\sqrt{-3+\sqrt{11}})}{4-\sqrt{11}}=$$$$-1+\sqrt{-3+\sqrt{11}}$$

Commented by MJS last updated on 01/Jun/20

$${\mathrm{you}}^{\prime }\mathrm{re}\mathrm{right}$$

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

$$\mathrm{thanks}\mathrm{you}\mathrm{sir}$$

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

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