\((37)^{2}-(13)^{2}\)
= (37 + 13) (37 - 13)
= \(50 \times 24\)
= 1200

\((2 . 06)^{2}-(0 . 94)^{2}\)
= (2.06 + 0.94) (2.06 - 0.94)
= (3.00) (1.12)
= 3.36

\(78 \times 82\)
= (80 -2) (80 + 2)
= \((80)^{2}-(2)^{2}\)
= 6400 - 4
= 6396

\(1 . 15 \times 0 . 85\)
\(=(1+. 15)(1-. 15)\)
\(=1^{2}-(.15)^{2}\)
= 1- 00225
= 0.9775

\((65)^{2}-(35)^{2}=(65+35)(65-35)\)
= \(100 \times 30=3000\)

\(k-p^{2}=(9+p)(9-p)\)
\(k-p^{2}=9^{2}-p^{2}\)
\(k=9^{2}-p^{2}+p^{2}\)
\(k=81\)

\(25-4 x^{2}=(5+a x)(5-a x)\)
\(25-4 x^{2}=(5)^{2}-(a x)^{2}\)
\(25-a^{2} x^{2}=25-4 x^{2}\)
\(-a^{2} x^{2}=25-4 x^{2}-25\)
\(a^{2}=\frac{-4 x^{2}}{-x^{2}}=4\)
বা, \(a=\sqrt{4}=2\)

\((4-x) \times 4+x=\left(16-x^{2}\right)\)

\(25l^{2}-16 m^{2}\)

\(=(5l)^{2}-(4 m)^{2}\)

\(=(5l+4 m)(5l-4 m)\)

\(= (5l+ 4m) (5l-4m)\)

\(49 x^{4}-36 y^{4}\)
\(=\left(7 x^{2}\right)^{2}-\left(6 y^{2}\right)^{2}\)
\(=\left(7 x^{2}+6 y^{2}\right)\left(7 x^{2}-6 y^{2}\right)\)

\((2 a+b)^{2}-(a+b)^{2}\)
\(=(2 a+b+a+b)(2 a+b-a-b)\)
\(=(3 a+2 b)(a)=(3 a+2 b) a\)

\((x+y)^{2}-(a+b)^{2}\)
\(=(x+y+a+b)(x+y-a-b)\)

\((x+y-z)^{2}-(x-y+z)^{2}\)
\(=(x+y-z+x-y+z)(x+y-z-x+y-z)\)
\(=2 x(2 y-2 z)=2 x \cdot 2(y-z)=4 x(y-z)\)

\((m+p+q)^{2}-(m-p-q)^{2}\)
\(=(m+p+q+m-p-q)(m+p+q-m+p+q)\)
\(=2 m(2 p+2 q)=2 m \cdot 2(p+q)\)
\(=4 m(p+q)\)

\((c+d)(c-d)\left(c^{2}+d^{2}\right)\)
\(=\left(c^{2}-d^{2}\right)\left(c^{2}+d^{2}\right)\)
\(=\left(c^{2}\right)^{2}-\left(d^{2}\right)^{2}\)
\(=c^{4}-d^{4}\)

\(\left(1-3 x^{2}\right)\left(1+3 x^{2}\right)\left(1+9 x^{4}\right)\)
\(=\left\{(1)^{2}-\left(3 x^{2}\right)^{2}\right\}\left(1+9 x^{4}\right)\)
\(=\left(1-9 x^{4}\right)\left(1+9 x^{4}\right)\)
\(=(1)^{2}-\left(9 x^{4}\right)^{2}\)
\(=1-81 x^{8}\)

\(\left(a^{2}+b^{2}\right)\left(a^{2}-b^{2}\right)\left(a^{4}+b^{4}\right)\left(a^{8}+b^{8}\right)\)
\(=\left(a^{4}-b^{4}\right)\left(a^{4}+b^{4}\right)\left(a^{8}+b^{8}\right)\)
\(=\left(a^{8}-b^{8}\right)\left(a^{8}+b^{8}\right)\)
\(=\left(a^{8}\right)^{2}-\left(b^{8}\right)^{2}=a^{16}-b^{16}\)

\(16 c^{4}-81 d^{4}\)
\(=\left(4 c^{2}\right)^{2}-\left(9 d^{2}\right)^{2}\)
\(=\left(4 c^{2}-9 d^{2}\right)\left(4 c^{2}+9 d^{2}\right)\)
\(=\left\{(2 c)^{2}-(3 d)^{2}\right\}\left(4 c^{2}+9 d^{2}\right)\)
\(=(2 c+3 d)(2 c-3 d)\left(4 c^{2}+9 d^{2}\right)\)

\({p}^{4} {q}^{4}-{r}^{4} {s}^{4}\)
\(=\left({p}^{2} {q}^{2}\right)^{2}-\left({r}^{2} {s}^{2}\right)^{2}\)
\(=\left({p}^{2} {q}^{2}+{r}^{2} {s}^{2}\right)\left({p}^{2} {q}^{2}-{r}^{2} {s}^{2}\right)\)
\(=\left({p}^{2} {q}^{2}+{r}^{2} {s}^{2}\right)\left\{({pq})^{2}-({rs})^{2}\right\}\)
\(=\left({p}^{2} {q}^{2}+r^{2} {s}^{2}\right)({pq}+{rs})({pq}-{rs})\)

\(81-x^{4}\)
\(=(9)^{2}-\left(x^{2}\right)^{2}\)
\(=\left(9+x^{2}\right)\left(9-x^{2}\right)\)
\(=\left(9+x^{2}\right)\left\{(3)^{2}-(x)^{2}\right\}\)
\(=\left(9+x^{2}\right)(3 +x)(3 -x)\)

\(625-a^{4} b^{4}\)
\(=(25)^{2}-\left(a^{2} b^{2}\right)^{2}\)
\(=\left(25+a^{2} b^{2}\right)\left(25-a^{2} b^{2}\right)\)
\(=\left(25+a^{2} b^{2}\right)\left\{(5)^{2}-(a b)^{2}\right\}\)
\(=\left(25+a^{2} b^{2}\right)(5+a b)(5-a b)\)

\(\mathbf{L} \cdot \mathbf{H} \cdot \mathbf{S}\left\{(p+q)^{2}\right\}^{2}-\left\{(p-q)^{2}\right\}^{2}\)
\(=\left\{(p+q)^{2}+(p-q)^{2}\right\}\left\{(p+q)^{2}-(p-q)^{2}\right\}\)
\(=2\left(p^{2}+q^{2}\right) \cdot 4 p q\)
\(=8 p q\left(p^{2}+q^{2}\right)=R \cdot H \cdot S\)
\(\left[(a+b)^{2}+(a-b)^{2}=2\left(a^{2}+b^{2}\right)\right.\)
\((a+b)^{2}-(a-b)^{2}=4 a b\) সুত্রদুটি প্রয়োগ করে]

\((a+b+c)(b+c-a)(c+a-b)(a+b-c)\)
\(=\{(b+c)+a)\}\{(b+c)-a\}\{a-(b-c)\}\{a+(b-c)\}\)
\(=\left\{(b+c)^{2}-(a)^{2}\right\}\left\{(a)^{2}-(b-c)^{2}\right\}\)
\(=2 a^{2} b^{2}+2 b^{2} c^{2}+2 c^{2} a^{2}-a^{4}-b^{4}-c^{4}\)

\(x=\frac{a}{b}+\frac{b}{a}=\frac{a^{2}+b^{2}}{a b}, y=\frac{a}{b}-\frac{b}{a}=\frac{a^{2}-b^{2}}{a b}\)
বামপক্ষ = \(x^{4}+y^{4}-2 x^{2} y^{2}=\left(x^{2}-y^{2}\right)^{2}\)
\(=\left\{\left(\frac{a^{2}+b^{2}}{a b}\right)^{2}-\left(\frac{a^{2}-b^{2}}{a b}\right)^{2}\right\}^{2}\)
\(=\left\{\frac{a^{4}+b^{4}+2 a^{2} b^{2}}{a^{2} b^{2}}-\frac{a^{4}+b^{4}-2 a^{2} b^{2}}{a^{2} b^{2}}\right\}^{2}\)
\(=\left\{\frac{a^{4}+b^{4}+2 a^{2} b^{2}-a^{4}-b^{4}+2 a^{2} b^{2}}{a^{2} b^{2}}\right\}^{2}\)
\(=\left(\frac{4 a^{2} b^{2}}{a^{2} b^{2}}\right)^{2}\)
\(=(4)^{2}=16\) = ডানপক্ষ

\(\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)\left(a^{4}-a^{2}+1\right)\)
\(=\left(a^{2}+1+a\right)\left(a^{2}+1-a\right)\left(a^{4}+1-a^{2}\right)\)
\(=\left\{\left(a^{2}+1\right)^{2}-(a)^{2}\right\}\left(a^{4}+1-a^{2}\right)\)
\(=\left(a^{4}+2 a^{2}+1-a^{2}\right)\left(a^{4}+1-a^{2}\right)\)
\(=\left(a^{4}+1+a^{2}\right)\left(a^{4}+1-a^{2}\right)\)
\(=\left(a^{4}+1\right)^{2}-\left(a^{2}\right)^{2}=a^{8}+2 a^{4}+1-a^{4}\)
\(=a^{8}+a^{4}+1\)

\(x^{4} y^{4}-2 x^{2} y^{2}=\left(x^{2}-y^{2}\right)^{2}\)
\(=\left\{\left(a+\frac{1}{a}\right)^{2}-\left(a-\frac{1}{a}\right)^{2}\right\}^{2}\)
\(=\left(4 \cdot a \cdot \frac{1}{a}\right)^{2}\left[a^{2}-b^{2}=4 a b\right]\) সূত্র প্রয়োগ করিয়া
\(=(4)^{2}=16\)

\(\left(4 x^{2}+4 x+1-a^{2}+8 a-16\right)\)
\(=(2 x)^{2}+2 \cdot 2 x \cdot 1+1^{2}-\left(a^{2}-8 a+16\right)\)
\(=(2 x+1)^{2}-\left\{(a)^{2}-2 \cdot a \cdot 4+(4)^{2}\right\}\)
\(=(2 x+1)^{2}-(a-4)^{2}\)

\(a^{2}+\frac{1}{a^{2}}-3\)
\(=a^{2}+\frac{1}{a^{2}}-2-1\)
\(=(a)^{2}-2 \cdot a \cdot \frac{1}{a}+\left(\frac{1}{a}\right)^{2}-(1)^{2}\)
\(=\left(a-\frac{1}{a}\right)^{2}-(1)^{2}\)