ঘনকের বাহুর দৈর্ঘ্য (একক)	ঘনকের আয়তন (ঘন একক)
(i)	\(p^{2}+q^{2}\)	\(\left(p^{2}\right)^{3}+3\left(p^{2}\right)^{2} \cdot q^{2}+3 \cdot p^{2} \times\left(q^{2}\right)^{2}+\left(q^{2}\right)^{3}\)
(ii)	\(\frac{x}{3}+\frac{4}{y}\)	\(\left(\frac{x}{3}\right)^{3}+3\left(\frac{x}{3}\right)^{2} \times \frac{4}{y}+3 \times \frac{x}{3} \times\left(\frac{4}{y}\right)^{2}+\left(\frac{4}{y}\right)^{3}\)
(iii)	\(x^{2} y-z^{2}\)	\(\left(x^{2} y^{2}\right)^{3}+3\left(x^{2} y^{2}\right)^{2} z^{2}+3 x^{2} y^{2}\left(z^{2}\right)^{2}+\left(z^{2}\right)^{3}\)
(iv)	\(1+b-2 c\)	\((1+b)^{3}-3(1+b)^{2} 2 c+3(1+b)(2 c)^{2}-(2 c)^{3}\)
(v)	5	\((2 \cdot 89)^{3}+(2 \cdot 11)^{3}+15 \times 2 \cdot 89 \times 2 \cdot 11\)
(vi)	\((2 m+3 n)(2 m-3 n)\)	\((2 m+3 n)^{3}+(2 m-3 n)^{3}+12 m\left(4 m^{2}-9 n^{2}\right)\)
(vii)	\((a+b)(a-b)\)	\((a+b)^{3}-(a-b)^{3}-6 b\left(a^{2}-b^{2}\right)\)
(viii)	\(2 x-3 y-4 z\)	\((2 x-3 y)^{3}-(4 z)^{3}-12 x(2 x-3 y)(2 x-3 y+4 z)\)
(ix)	\(x-5\)	\(x^{6}-15 x^{4}+75 x^{2}-125\)
(x)	\(10+x\)	\(1000+30 x(10+x)+x^{3}\)

(i) \( p^{2}+q^{2} \)বাহুবিশিষ্ট ঘনকের আয়তন—
\( \left(p^{2}+q^{2}\right)^{3}=\left(p^{2}\right)^{3}+3\left(p^{2}\right)^{2} \cdot q^{2}+3 p^{2}\left(q^{2}\right)^{2}+\left(q^{2}\right)^{3} \)
\( =p^{6}+3 p^{4} q^{2}+3 p^{2} q^{4}+q^{6} \)
(ii) \( \frac{x}{3}+\frac{4}{y} \)বাহুবিশিষ্ট ঘনকের আয়তন—
\( \left(\frac{x}{3}+\frac{4}{y}\right)^{3}=\left(\frac{x}{3}\right)^{3}+3 \cdot\left(\frac{x}{3}\right)^{2} \cdot \frac{4}{y}+3 \cdot \frac{x}{3} \cdot\left(\frac{4}{y}\right)^{2}+\left(\frac{4}{y}\right)^{3} \)
\( =\frac{x^{3}}{27}+3 \cdot \frac{x^{2}}{9 } \cdot \frac{4}{y}+x \cdot \frac{16}{y^{2}}+\frac{64}{y^{3}}\)
\(=\frac{x^{3}}{27}+\frac{4 x^{2}}{3 y}+\frac{16 x}{y^{2}}+\frac{64}{y^{3}} \)
(iii) \( x^{2} y-z^{2} \) বাহুবিশিষ্ট ঘনকের আয়তন—
\( \left(x^{2} y-z^{2}\right)^{3}=\left(x^{2} y\right)^{3}-3\left(x^{2} y\right)^{2}\left(z^{2}\right)+3 \cdot x^{2} y\left(z^{2}\right)^{2}-\left(z^{2}\right)^{3} \)
\( =x^{6} y^{3}-3 x^{4} y^{2} z^{2}+3 x^{2} y z^{4}-z^{6} \)
(iv) \( l+\mathrm{b}-2 \mathrm{c} \) বাহুবিশিষ্ট ঘনকের আয়তন—
\( (l+b-2 c)^{3}\)
\(=[(l+b)-2 c]^{3}\)
\(=(l+b)^{3}-3(l+b)^{2} \cdot 2 \mathrm{c}+3 \cdot(l+b) \cdot(2 c)^{2}-(2 c)^{3}\)
\(=l^{3}+3 l^{2} \mathrm{~b}+3 \mathrm{~b}^{2} l+\mathrm{b}^{3}-6 \mathrm{c}\left(l^{2}+2 l \mathrm{~b}+\mathrm{b}^{2}\right) +12 c^{2}(l+b)-8 c^{3} \)
\( =l^{3}+3 l^{2} \mathrm{~b}+3 \mathrm{~b}^{2} l+\mathrm{b}^{3}-6 \mathrm{c} l^{2}-12 \mathrm{bcl}-6 \mathrm{~b}^{2} \mathrm{c} +12 c^{2} l+12 b c^{2}-8 c^{3} \)
(v) \( (2.89)^{3}+(2.11)^{3}+15 \times 2.89 \times 2.11 \) আয়তনবিশিষ্ট ঘনকের বাহুর দৈর্ঘ্য
\( \sqrt[3]{(2.89)^{3}+(2.11)^{3}+15 \times 2.89 \times 2.11}\)
\(=\sqrt[3]{(2.89)^{3}+(2.11)^{3}+3 \times 2.89 \times 2.11 \times 5}\)
\(=\sqrt[3]{(2.89)^{3}+(2.11)^{3}+3 \times 2.89 \times 2.11 \times(2.89+2.11)}\)
\(=\sqrt[3]{(2.89+2.11)^{3}}=\sqrt[3]{(5)^{3}}=5 \)
(vi) \( (2 m+3 n)^{3}+(2 m-3 n)^{3}+12 m\left(4 m^{2}-9 n^{2}\right) \)
আয়তনবিশিষ্ট ঘনকের বাহুর দৈর্ঘ্য
\( \sqrt[3]{(2 m+3 n)^{3}+(2 m-3 n)^{3}+12 m\left(4 m^{2}-9 n^{2}\right)}\)
\(=\sqrt[3]{(2 m+3 n)^{3}+(2 m-3 n)^{3}+3 \times(2 m+3 n) \times(2 m-3 n) \times 4 m}\)
\(=\sqrt[3]{(2 m+3 n)^{3}+(2 m-3 n)^{3}+3 \times(2 m+3 n)(2 m-3 n) \times} \{(2 m+3 n)+(2 m-3 n)\} \)
\( =\sqrt[3]{(2 m+3 n+2 m-3 n)^{3}}=\sqrt[3]{(4 m)^{3}}=4 m \)
(vii) \( (a+b)^{3}-(a-b)^{3}-6 b\left(a^{2}-b^{2}\right) \) আয়তনবিশিষ্ট ঘনকের বাহুর দৈর্ঘ্য
\( \sqrt[3]{(a+b)^{3}-(a-b)^{3}-6 b\left(a^{2}-b^{2}\right)}\)
\(=\sqrt[3]{(a+b)^{3}-(a-b)^{3}-3(a+b)(a-b) \cdot 2 b}\)
\(=\sqrt[3]{(a+b)^{3}-(a-b)^{3}-3(a+b)(a-b)\{(a+b)-(a-b)\}}\)
\(=\sqrt[3]{\{(a+b)-(a-b)\}^{3}}\)
\(=\sqrt[3]{(a+b-a+b)^{3}}=\sqrt[3]{(2 b)^{3}}=2 b \)
(viii) \( 2 x-3 y-4 z \) বাহুবিশিষ্ট ঘনকের আয়তন—
\( (2 x-3 y-4 z)^{3}=\{(2 x-3 y)-4 z\}^{3}\)
\(=(2 x-3 y)^{3}-3 \cdot(2 x-3 y)^{2} \cdot 4 z+3 \cdot(2 x-3 y)(4 z)^{2}-(4 z)^{3} \)
\( =8 x^{3}-3 \cdot(2 x)^{2} \cdot 3 y+3 \cdot 2 x(3 y)^{2}-(3 y)^{3}-12 z\left(4 x^{2}-12 x y+9 y^{2}\right)+48 z^{2}(2 x-3 y)-64 z^{3} \)
\( =8 x^{3}-36 x^{2} y+54 x y^{2}-27 y^{3}-48 x^{2} z+144 x y z-108 y^{2} z+96 x z^{2}-144 y z^{2}-64 z^{3} \)
(ix) \( x^{6}-15 x^{4}+75 x^{2}-125 \) আয়তনবিশিষ্ট ঘনকের বাহুর দৈর্ঘ্য
\( \sqrt[3]{x^{6}-15 x^{4}+75 x^{2}-125}\)
\(=\sqrt[3]{\left(x^{2}\right)^{3}-3 \cdot\left(x^{2}\right)^{2} \cdot 5+3 \cdot x^{2} \cdot(5)^{2}-(5)^{3}}\)
\(=\sqrt[3]{\left(x^{2}-5\right)^{3}}=\left(x^{2}-5\right) \)
(x) \( 1000+30 x(10+x)+x^{3} \)আয়তনবিশিষ্ট ঘনকের বাহুর দৈর্ঘ্য
\( \sqrt[3]{1000+30 x(10+x)+x^{3}}\)
\(=\sqrt[3]{1000+300 x+30 x^{2}+x^{3}}\)
\(=\sqrt[3]{(10)^{3}+3 \cdot(10)^{2} x+3 \cdot 10 \cdot(x)^{2}+(x)^{3}}\)
\(=\sqrt[3]{(10+x)^{3}}=(10+x) \)

প্রদত্ত রাশিমালা, \( x^{3}-y^{3}-6 x y \)
\( =(x)^{3}-(y)^{3}-3 x y \cdot 2=(x)^{3}-(y)^{3}-3 \cdot x y \cdot(x-y) \)\( [\because x-y=2] \)
\( =(x-y)^{3}\) \( [\left.(a-b)^{3}=a^{3}-b^{3}-3 a b(a-b)\right] \)
\(=(2)^{3}=8 \)

প্রদত্ত, \( a+b=-\frac{1}{3} \)
\( (a+b)^{3}=\left(-\frac{1}{3}\right)^{3} \) [উভয়পক্ষকে ঘন করে পাই]
বা, \( a^{3}+b^{3}+3 a b(a+b)=-\frac{1}{27} \)
[\( (a+b)^{3}=a^{3}+b^{3}+3 a b(a+b) \)]
বা, \( a^{3}+b^{3}+3 a b\left(-\frac{1}{3}\right)=\frac{-1}{27} \) \( \left[\because a+b=-\frac{1}{3}\right] \)
বা, \( a^{3}+b^{3}-a b=-\frac{1}{27} \) (প্রমাণিত)

প্রদত্ত, \( x+y=2 \)
এবং \( \frac{1}{x}+\frac{1}{y}=2 \)
বা, \( \frac{x+y}{x y}=2 \)
বা, \( \frac{2}{x y}=2[\because x+y=2] \)
বা, \( 2 x y=2\)
বা, \(x y=\frac{2}{2}=1 \)
\( \therefore x^{3}+y^{3}=(x+y)^{3}-3 x y(x+y) \)
[\( a^{3}+b^{3}=(a+b)^{3}-3 a b(a+b) \)[\( a^{3}+b^{3}=(a+b)^{3}-3 a b(a+b) \) সূত্র থেকে পাই]
\( =(2)^{3}-3.1 .2 \) [\( \because x+y=2 \) এবং \( x y=1 \)]
\( =8-6=2\)
\(\therefore\) \(x^{3}+y^{3}=2 \)

প্রদত্ত, \( \frac{x^{2}-1}{x}=2 \)
বা, \( \frac{x^{2}}{x}-\frac{1}{x}=2 \) বা, \( x-\frac{1}{x}=2 \)
এখন, \( \frac{x^{6}-1}{x^{3}}=\frac{x^{6}}{x^{3}}-\frac{1}{x^{3}}=x^{3}-\frac{1}{x^{3}} \)
\( =\left(x-\frac{1}{x}\right)^{3}+3 \cdot x \cdot \frac{1}{x}\left(x-\frac{1}{x}\right) \)
[\( a^{3}-b^{3}=(a-b)^{3}+3 a b(a-b) \)]
\( =(2)^{3}+3 \times 2\) \(\quad \left[\because x-\frac{1}{x}=2\right] \)
\(=8+6=14 \)
\( \therefore \frac{x^{6}-1}{x^{3}}=14 \)

প্রদত্ত \( x+\frac{1}{x}=5 \)
\( \left(x+\frac{1}{x}\right)^{3}=(5)^{3} \) [ উভয়পক্ষকে ঘন করে পাই]
\( \therefore x^{3}+\frac{1}{x^{3}}+3 \cdot x \cdot \frac{1}{x}\left(x+\frac{1}{x}\right)=125 \)
[\( (a+b)^{3}=a^{3}+b^{3}+3 a b(a+b) \)]
বা, \( x^{3}+\frac{1}{x^{3}}+3.5=125\left[\because x+\frac{1}{x}=5\right]\)
বা, \(x^{3}+\frac{1}{x^{3}}=125-15=110\)
\(\therefore\) \(x^{3}+\frac{1}{x^{3}}=110 \)

প্রদত্ত, \( x=y+z \)
বা, \( x-y=z\)
\((x-y)^{3}=z^{3} \) [উভয়পক্ষকে ঘন করে পাই]
বা, \( x^{3}-y^{3}-3 x y(x-y)=z^{3} \)
[ \( (a-b)^{3}=a^{3}-b^{3}-3 a b(a-b) \) ]
বা, \( x^{3}-y^{3}-3 x y z=z^{3} \quad[\because x-y=z]\)
বা, \(x^{3}-y^{3}-z^{3}-3 x y z=0 \)

প্রদত্ত, \( x y(x+y)=m \quad \) বা, \( x+y=\frac{m}{x y} \)
বামপক্ষ \( =x^{3}+y^{3}+3 m\)
\(=x^{3}+y^{3}+3 x y(x+y) \quad[\because m=x y(x+y)]\)
\(=(x+y)^{3} \)
[\( (a+b)^{3}=a^{3}+b^{3}+3 a b(a+b) \)]
\( =\left(\frac{m}{x y}\right)^{3}\left[\because x+y=\frac{m}{x y}\right] \)
\( =\frac{\mathrm{m}^{3}}{x^{3} \mathrm{y}^{3}}= \) ডানপক্ষ (প্রমাণিত)

প্রদত্ত, \( 2 x+\frac{1}{3 x}=4 \)
উভয়ক্ষকে \( \frac{3}{2} \) দিয়ে গুণ করে পাই,
\( \left(2 x+\frac{1}{3 x}\right) \times \frac{3}{2}=4 \times \frac{3}{2} \)
বা, \( 2 x \times \frac{3}{2}+\frac{1}{3 x} \times \frac{3}{2}=6 \)
বা, \( 3 x+\frac{1}{2 x}=6 \)
বামপক্ষ, \( 27 x^{3}+\frac{1}{8 x^{3}}=(3 x)^{3}+\left(\frac{1}{2 x}\right)^{3} \)
\( =\left(3 x+\frac{1}{2 x}\right)^{3}-3 \cdot 3 x \cdot \frac{1}{2 x}\left(3 x+\frac{1}{2 x}\right) \)
[\( a^{3}+b^{3}=(a+b)^{3}-3 a b(a+b) \) সূত্র থেকে পাই]
\( \quad\left[a^{3}+b^{3}=(a+b)^{3}-3 a b(a+b)\right.\)
\(=(6)^{3}-\left(\frac{9}{2} \times 6^{3}\right) \quad\left[\because 3 x+\frac{1}{2 x}=6\right] \)
\(=216-27=189\)
=ডানপক্ষ (প্রমাণিত)

প্রদত্ত,
\( 2 a-\frac{2}{a}+1=0 \)
বা, \( 2 a-\frac{2}{a}=-1\)
বা, \(2\left(a-\frac{1}{a}\right)=-1\)
বা, \(a-\frac{1}{a}=-\frac{1}{2}\)
\(\therefore\) \(a^{3}-\frac{1}{a^{3}}+2 \)
\( =\left(a-\frac{1}{a}\right)^{3}+3 \cdot a \cdot \frac{1}{a}\left(a-\frac{1}{a}\right)+2 \)
[\( a^{3}-b^{3}=(a-b)^{3}+3 a b(a-b) \)
\( =\left(-\frac{1}{2}\right)^{3}+3\left(-\frac{1}{2}\right)+2\quad \left[\because a-\frac{1}{a}=-\frac{1}{2}\right]\)
\(=-\frac{1}{8}-\frac{3}{2}+2=\frac{-1-12+16}{8}=\frac{3}{8}\)
\(\therefore\)\(a^{3}-\frac{1}{a^{3}}+2=\frac{3}{8} \)

\( a^{3}+b^{3}+c^{3}=3 a b c\)
বা, \(a^{3}+b^{3}+c^{3}-3 a b c=0\)
বা, \((a+b)^{3}-3 a b(a+b)+c^{3}-3 a b c=0 \)
[\( a^{3}+b^{3}=(a+b)^{3}-3 a b(a+b) \) সূত্র থেকে পাই]
বা, \( {\left[(a+b)^{3}+(c)^{3}\right]-3 a b(a+b+c)=0}\)
বা, \((a+b+c)^{3}-3(a+b) \cdot c(a+b+c)-3 a b(a+b+c)=0 \)
[\( a^{3}+b^{3}=(a+b)^{3}-3 a b(a+b) \)সূত্র থেকে পাই]
বা, \( (a+b+c)\left\{(a+b+c)^{2}-3 c(a+b)-3 a b\right\}=0 \)
দুটি রাশির গুণফল শূন্য হলে, তাদের মধ্যে যে-কোনো একটি রাশির মান শূন্য হবে।
হয়, \( \left\{(a+b+c)^{2}-3 c(a+b)-3 a b\right\}=0 \)
বা, \( a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a-3 a c-3 b c-3 a b=0\)
বা, \(a^{2}+b^{2}+c^{2}-a b-b c-c a=0\)
বা, \(2\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)=2 \times 0\)
বা, \(2 a^{2}+2 b^{2}+2 c^{2}-2 a b-2 b c-2 c a=0\)
বা, \(a^{2}-2 a b+b^{2}+b^{2}-2 b c+c^{2}+c^{2}-2 c a+a^{2}=0\)
বা, \((a-b)^{2}+(b-c)^{2}+(c-a)^{2}=0 \)
তিনটি বর্গ রাশির সমষ্টি শূন্য হলে, ওদের মান পৃথক পৃথক ভাবে শূন্য হবে।
\(\therefore\) \( (a-b)^{2}=0 \)
বা, \( a-b=0\)
\(\therefore\) \(a=b\)
\(\therefore\) \(a=b=c \)
\(\therefore\) \( (b-c)^{2}=0\)
বা, \(b-c=0\)
\(\therefore\) \(b=c \)
\(\therefore\) \( (c-a)^{2}=0\)
বা, \(c-a=0\)
\(\therefore\) \(c=a \)
কিন্তু, যেহেতু \( a \neq b \neq c \)
\( \therefore \left\{(a+b+c)^{2}-3 c(a+b)-3 a b\right\} \neq 0\)
\(\therefore a+b+c=0 \)

প্রদত্ত, \( m+n=5 \) এবং \( m n=6 \)
\(\therefore\) \( m^{2}+n^{2}=(m+n)^{2}-2 m n \)
\(\quad \quad \quad =(5)^{2}-2.6 \quad[\because m+n=5\) ও \( m n=6]\)
\(\quad \quad \quad \quad =25-12=13 \)
আবার, \( m^{3}+n^{3}=(m+n)^{3}-3 m n(m+n) \)
[\( a^{3}+b^{3}=(a+b)^{3}-3 a b(a+b) \)সূত্র থেকে পাই]
\( \quad \quad \quad \quad =(5)^{3}-3.6 .5 \)
\( \quad \quad \quad \quad \quad [\because m+n=5\) ও \( m n=6 \)
\( =125-90=35 \)
\(\therefore\)\( \left(m^{2}+n^{2}\right)\left(m^{3}+n^{3}\right)=13 \times 35=455 \)