Evaluate (3+2)^ 6 -(3-2)^ 6 .

Question

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

✓

Answer

Using Binomial Theorem.
This can be done as
$(a+b)^6={ }^6 C_0 a^6+{ }^6 C_1 a^5 b+{ }^6 C_2 a^4 b^2+{ }^6 C_3 a^3 b^3+{ }^6 C_4 a^2 b^4+{ }^6 C_5 a^1 b^5+{ }^6 C_6 b^6$
$=a^6+6 a^5 b+15 a^4 b^2+20 a^3 b^3+15 a^2 b^4+6 a b^5+b^6$
$(a-b)^6={ }^6 C_0 a^6-{ }^6 C_1 a^5 b+{ }^6 C_2 a^4 b^2-{ }^6 C_3 a^3 b^3+{ }^6 C_4 a^2 b^4-{ }^6 C_5 a^1 b^5+{ }^6 C_6 b^6$
$=a^6-6 a^5 b+15 a^4 b^2-20 a^3 b^3+15 a^2 b^4-6 a b^5+b^6$
Therefore $(a+b)^6-(a-b)^6=2\left[6 a^5 b+20 a^3 b^3+6 a b^5\right]$
Puting a = $\sqrt3$ and b = $\sqrt2$, we obtain
$(\sqrt{3}+\sqrt{2})^{6}-(\sqrt{3}-\sqrt{2})^{6}=2\left[6(\sqrt{3})^{5}(\sqrt{2})+20(\sqrt{3})^{3}(\sqrt{2})^{3}+6(\sqrt{3})(\sqrt{2})^{5}\right]$
$=2[54 \sqrt{6}+120 \sqrt{6}+24 \sqrt{6}]$
$=2 \times 198 \sqrt{6}$
$=396 \sqrt{6}$

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