Evaluate the following: (2+1)^6+(2-1)^6 - Vidyadip

Question

Evaluate the following:

$(\sqrt{2}+1)^6+(\sqrt{2}-1)^6$

✓

Answer

$(\sqrt{2}+1)^6+(\sqrt{2}-1)^6$

$={^6\text{C}}_0(\sqrt2)^6+{^6\text{C}}_1(\sqrt2)^5+{^6\text{C}}_2(\sqrt2)^4+{^6\text{C}}_3(\sqrt2)^3\\{^6\text{C}}_4(\sqrt2)^2+{^6\text{C}}_5(\sqrt2)+{^6\text{C}}_6+{^6\text{C}}_0(\sqrt2)^6-\\{^6\text{C}}_1(\sqrt2)^5+{^6\text{C}}_2(\sqrt2)^4-{^6\text{C}}_3(\sqrt2)^3+{^6\text{C}}_3(\sqrt2)^3-{^6\text{C}}_4(\sqrt2)^2-{^6\text{C}}_6(\sqrt2)^0$

$=2\big[2^3+15\times2^2+15\times2+1\big]$

$=2\big[8+60+30+1\big]=2(99)=198$

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