Select and use the identity that will help you to find the following products without multiplying directly:

(A) $(41)^2$
(B) $(27)^2$
(C) $(23 \times 17)$
(D) $(135)^2$
(E) $(97)^2$
(F) $(18 \times 29)$
(G) $(34 \times 43)$
(H) $(205)^2$                       [Page No. 81]

(A) $(40 + 1)^2 = 1600 + 80 + 1 = 1681$

(B) $(30 - 3)^2 = 900 - 180 + 9 = 729$

(C) $23 \times 17 = (20 + 3)(20 - 3)<span class="katex-error" title="ParseError: KaTeX parse error: Can't use function '$' in math mode at position 1: ̲\begin{aligned}…" style="color:#cc0000">\begin{aligned} &= 20^2 - 3^2 \\ &= 400 - 9 = 391 \end{aligned}

(D) $(100 + 30 + 5)^2$
$$\begin{aligned} &= 10000 + 900 + 25 + 6000 + 300 + 1000 \\ &= 18225 \end{aligned}$$

(E) $(100 - 3)^2 = 10000 - 600 + 9 = 9409$

(F) $(18 + 29)^2 = 18^2 + 2(18)(29) + 29^2$
$$\begin{aligned} 47^2 &= 324 + 2(18)(29) + 841 \\ 2209 &= 324 + 2(18)(29) + 841 \\ 2209 &= 1165 + 2(18)(29) \end{aligned}$$
$$\begin{aligned} 2209 - 1165 &= 2(18)(29) \\ 1044 &= 2(18)(29) \\ 18 \times 29 &= \frac{1044}{2} \\ 18 \times 29 &= 522 \end{aligned}$$
Hence, $18 \times 29 = 522$

$$\begin{aligned} (G) \quad (34 + 43)^2 &= 34^2 + 2(34)(43) + 43^2 \\ 77^2 &= 1156 + 2(34)(43) + 1849 \\ 5929 &= 3005 + 2(34)(43) \\ 5929 - 3005 &= 2(34)(43) \\ 2924 &= 2(34)(43) \\ 34 \times 43 &= \frac{2924}{2} \\ 34 \times 43 &= 1462 \end{aligned}$$
Hence, $34 \times 43 = 1462$ .

$$(H)\quad(200+5)^2=40000+2000+2=42025$$