We have (√(3)+√(2))^6 (√(3) √(2))^6 =[6_C_0 (√(3))^6 + 6_C_1(√(3))^5.√(2) + 6_C_2(√(3))^4(√(2))^2+ 6_C_3(√(3))^3(√(2))^3+6_C_4(√(3))^2(√(2))^4+ 6_C_5(√(3 ...

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Evaluate √3+√...

Question

Evaluate $(\sqrt{3} + \sqrt{2})^{6} - (\sqrt{3} - \sqrt{2})^{6}$ .

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Solution

We have $(\sqrt{3} + \sqrt{2})^{6} - (\sqrt{3} - \sqrt{2})^{6}$

$= [6_{C_{0}} (\sqrt{3})^{6} + 6_{C_{1}} (\sqrt{3})^{5} . \sqrt{2} + 6_{C_{2}} (\sqrt{3})^{4} (\sqrt{2})^{2} + 6_{C_{3}} (\sqrt{3})^{3} (\sqrt{2})^{3} + 6_{C_{4}} (\sqrt{3})^{2} (\sqrt{2})^{4} + 6_{C_{5}} (\sqrt{3}) (\sqrt{2})^{5} + 6_{C_{6}} (\sqrt{2})^{6}] - [6_{C_{0}} (\sqrt{3})^{6} - 6_{C_{1}} (\sqrt{3})^{5} \sqrt{2} + 6_{C_{2}} (\sqrt{3})^{4} (\sqrt{2})^{2} - 6_{C_{3}} (\sqrt{3})^{3} (\sqrt{2})^{3} + 6_{C_{4}} (\sqrt{3})^{2} (\sqrt{2})^{4} - 6_{C_{5}} (\sqrt{3}) (\sqrt{2})^{5} + 6_{C_{6}} (\sqrt{2})^{6}]$

$= 6_{C_{0}} (\sqrt{3})^{6} + 6_{C_{1}} (\sqrt{3})^{5} \sqrt{2} + 6_{C_{2}} (\sqrt{3})^{4} (\sqrt{2})^{2} + 6_{C_{3}} (\sqrt{3})^{3} (\sqrt{2})^{3} + 6_{C_{4}} (\sqrt{3})^{2} (\sqrt{2})^{4} + 6_{C_{5}} (\sqrt{3}) (\sqrt{2})^{5} + 6_{C_{6}} (\sqrt{2})^{6} - 6_{C_{0}} (\sqrt{3})^{6} + 6_{C_{1}} (\sqrt{3})^{5} \sqrt{2} - 6_{C_{2}} (\sqrt{3})^{4} (\sqrt{2})^{2} + 6_{C_{3}} (\sqrt{3})^{3} (\sqrt{2})^{3} - 6_{C_{4}} (\sqrt{3})^{2} (\sqrt{2})^{4} + 6_{C_{5}} (\sqrt{3}) (\sqrt{2})^{5} - 6_{C_{6}} (\sqrt{2})^{6}$

$= 2 \times 6_{C_{1}} (\sqrt{3})^{5} . \sqrt{2} + 2 \times 6_{C_{3}} (\sqrt{3})^{3} (\sqrt{2})^{3} + 2 \times 6_{C_{5}} (\sqrt{3}) (\sqrt{2})^{5}$

$= (2 \times 6 \times 9 \sqrt{3} \times \sqrt{2}) + (2 \times 20 \times 3 \sqrt{3} \times 2 \sqrt{3}) + (2 \times 6 \times \sqrt{3} \times 4 \sqrt{2})$

$= 108 \sqrt{6} + 240 \sqrt{6} + 48 \sqrt{6} = 396 \sqrt{6}$

Suggest Corrections

Evaluate (√3+√2)6−(√3−√2)6.

Evaluate $(\sqrt{3} + \sqrt{2})^{6} - (\sqrt{3} - \sqrt{2})^{6}$ .