# Evaluate :

Question:

Evaluate :

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

Solution:

To find: Value of $(\sqrt{2}+1)^{6}+(\sqrt{2}-1)^{6}$

Formula used: (i) ${ }^{n} C_{r}=\frac{n !}{(n-r) !(r) !}$

(ii) $(a+b)^{n}={ }^{n} C_{0} a^{n}+{ }^{n} C_{1} a^{n-1} b+{ }^{n} C_{2} a^{n-2} b^{2}+\ldots \ldots+{ }^{n} C_{n-1} a b^{n-1}+{ }^{n} C_{n} b^{n}$

$(a+1)^{6}=$

$\left[{ }^{6} \mathrm{C}_{0} \mathrm{a}^{6}\right]+\left[{ }^{6} \mathrm{C}_{1} \mathrm{a}^{6-1} 1\right]+\left[{ }^{6} \mathrm{C}_{2} \mathrm{a}^{6-2} 1^{2}\right]+\left[{ }^{6} \mathrm{C}_{3} \mathrm{a}^{6-3} 1^{3}\right]+\left[{ }^{6} \mathrm{C}_{4} \mathrm{a}^{6-4} 1^{4}\right]+$

$\left[{ }^{6} \mathrm{C}_{5} \mathrm{a}^{6-5} 1^{5}\right]+\left[{ }^{6} \mathrm{C}_{6} 1^{6}\right]$

$\Rightarrow{ }^{6} \mathrm{C} 0 \mathrm{a}^{6}+{ }^{6} \mathrm{C}_{1} \mathrm{a}^{5}+{ }^{6} \mathrm{C} 2 \mathrm{a}^{4}+{ }^{6} \mathrm{C} 3 \mathrm{a}^{3}+{ }^{6} \mathrm{C} 4 \mathrm{a}^{2}+{ }^{6} \mathrm{C} 5 \mathrm{a}+{ }^{6} \mathrm{C} 6 \ldots$ (i)

$(a-1)^{6}=$

$\left[{ }^{6} \mathrm{C}_{0} \mathrm{a}^{6}\right]+\left[{ }^{6} \mathrm{C}_{1} \mathrm{a}^{6-1}(-1)^{1}\right]+\left[{ }^{6} \mathrm{C}_{2} \mathrm{a}^{6-2}(-1)^{2}\right]+\left[{ }^{6} \mathrm{C}_{3} \mathrm{a}^{6-3}(-1)^{3}\right]+$

$\left[{ }^{6} \mathrm{C}_{4} \mathrm{a}^{6-4}(-1)^{4}\right]+\left[{ }^{6} \mathrm{C}_{5} \mathrm{a}^{6-5}(-1)^{5}\right]+\left[{ }^{6} \mathrm{C}_{6}(-1)^{6}\right]$

$\Rightarrow{ }^{6} \mathrm{C} 0 \mathrm{a}^{6}-{ }^{6} \mathrm{C} 1 \mathrm{a}^{5}+{ }^{6} \mathrm{C} 2 \mathrm{a}^{4}-{ }^{6} \mathrm{C} 3 \mathrm{a}^{3}+{ }^{6} \mathrm{C} 4 \mathrm{a}^{2}-{ }^{6} \mathrm{C} 5 \mathrm{a}+{ }^{6} \mathrm{C} 6 \ldots$ (ii)

Adding eqn. (i) and (ii)

$(\mathrm{a}+1)^{6}+(\mathrm{a}-1)^{6}=\left[{ }^{6} \mathrm{C}_{0} \mathrm{a}^{6}+{ }^{6} \mathrm{C}_{1} \mathrm{a}^{5}+{ }^{6} \mathrm{C}_{2} \mathrm{a}^{4}+{ }^{6} \mathrm{C}_{3} \mathrm{a}^{3}+{ }^{6} \mathrm{C}_{4} \mathrm{a}^{2}+{ }^{6} \mathrm{C}_{5} \mathrm{a}+{ }^{6} \mathrm{C}_{6}\right]+$

$\left[{ }^{6} \mathrm{C}_{0} \mathrm{a}^{6}-{ }^{6} \mathrm{C}_{1} \mathrm{a}^{5}+{ }^{6} \mathrm{C}_{2} \mathrm{a}^{4}-{ }^{6} \mathrm{C}_{3} \mathrm{a}^{3}+{ }^{6} \mathrm{C}_{4} \mathrm{a}^{2}-{ }^{6} \mathrm{C}_{5} \mathrm{a}+{ }^{6} \mathrm{C}_{6}\right]$

$\Rightarrow 2\left[{ }^{6} \mathrm{C} 0 \mathrm{a}^{6}+{ }^{6} \mathrm{C} 2 \mathrm{a}^{4}+{ }^{6} \mathrm{C} 4 \mathrm{a}^{2}+{ }^{6} \mathrm{C} 6\right]$

$\Rightarrow 2^{\left[\left(\frac{6 !}{0 !(6-0) !} a^{6}\right)+\left(\frac{6 !}{2 !(6-2) !} a^{4}\right)+\left(\frac{6 !}{4 !(6-4) !} a^{2}\right)+\left(\frac{6 !}{6 !(6-6) !}\right)\right]}$

$\Rightarrow 2\left[(1) a^{6}+(15) a^{4}+(15) a^{2}+(1)\right]$

$\Rightarrow 2\left[a^{6}+15 a^{4}+15 a^{2}+1\right]=(a+1)^{6}+(a-1)^{6}$

Putting the value of $a=\sqrt{2}$ in the above equation

$(\sqrt{2}+1)^{6}+(\sqrt{2}-1)^{6}=2\left[(\sqrt{2})_{6}+15(\sqrt{2})_{4}+15(\sqrt{2})_{2}+1\right]$

$\Rightarrow 2[8+15(4)+15(2)+1]$

$\Rightarrow 2[8+60+30+1]$

$\Rightarrow 2[99]$

$\Rightarrow 198$

Ans) 198