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Question Number 193026 by Mingma last updated on 02/Jun/23

Answered by aleks041103 last updated on 02/Jun/23

$${(3+\sqrt{5})}^3=3^3+3\times 3^2\times \sqrt{5}+3\times 3\times 5+5\sqrt{5}=$$$$=27+45+32\sqrt{5}=72+32\sqrt{5}$$$$32\sqrt{5}=\sqrt{5\times {32}^2}=\sqrt{5\times 1024}=\sqrt{5120}=71+r$$$$0<r<1$$$$andreally$$$${71}^2={(70+1)}^2=4900+1+140=5041$$$${72}^2={(70+2)}^2=4900+4+280=5184$$$${71}^2<5120<{72}^2\Rightarrow 71<\sqrt{5120}<72$$$$\Rightarrow 143<72+32\sqrt{5}<144$$$$\Rightarrow Ans.143$$

Commented by Mingma last updated on 02/Jun/23

Perfect ��

Answered by Alleddawi last updated on 02/Jun/23

$${(3+\sqrt{5})}^3=72+32\sqrt{5}$$$$32\sqrt{5}=\sqrt{1024\times 5}=\sqrt{5120}$$$$5041<5120<5184$$$$\sqrt{5041}<\sqrt{5120}<\sqrt{5184}$$$$71<\sqrt{5120}<72$$$$71<32\sqrt{5}<72$$$$72+71<72+32\sqrt{5}<72+72$$$$143<{(3+\sqrt{5})}^3<144$$$${(3+\sqrt{5})}^3\approx 143$$

Commented by Mingma last updated on 03/Jun/23

Perfect ��

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