Simplify each of the following:

Solution:

(i) We have

(ii)  We have,

$322 \times 322-2 \times 322 \times 22+22 \times 22$

$=(322-22)^{2} \quad\left[a^{2}+b^{2}-2 a b=(a-b)^{2}\right]$

$=(300)^{2} \quad[$ Where $a=322$ and $b=22]$

$=90000$

Therefore, 322 × 322 - 2 × 322 × 22 + 22 × 22 = 90000.

(iii)  We have,

0.76 × 0.76 + 2 × 0.76 × 0.24 + 0.24 × 0.24

$=(0.76+0.24)^{2} \quad\left[a^{2}+b^{2}+2 a b=(a+b)^{2}\right]$

$=(1.00)^{2} \quad[$ Where $a=0.76$ and $b=0.24]$

= 1

Therefore, 0.76 × 0.76 + 2 × 0.76 × 0.24 + 0.24 × 0.24 = 1.

(iv) We have,

$\frac{7.83 * 7.83-1.17 * 1.17}{6.66}$

$=\frac{(7.83+1.17)(7.83-1.17)}{6.66}\left[\therefore(a-b)^{2}=(a+b)(a-b)\right]$

$=\frac{(9.00)(6.66)}{6.66}=9$

$\therefore \frac{7.83 * 7.83-1.17 * 1.17}{6.66}=9$