Simplify each of the following:
(i) $175 \times 175+2 \times 175 \times 25+25 \times 25$
(ii) $322 \times 322-2 \times 322 \times 22+22 \times 22$
(iii) $0.76 \times 0.76+2 \times 0.76 \times 0.24+0.24 \times 0.24$
(iv) $\frac{7.83 * 7.83-1.17 * 1.17}{6.66}$
(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,
$\frac{7.83 * 7.83-1.17 * 1.17}{6.66}$
$=\frac{(7.83+1.17)(7.83-1.17)}{6.66}\left[\therefore(a-b)^{2}=(a+b)(a-b)\right]$
$=\frac{(9.00)(6.66)}{6.66}=9$
$\therefore \frac{7.83 * 7.83-1.17 * 1.17}{6.66}=9$
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