সূত্রের সাহায্যে নিচের রাশিগুলোর ঘন নির্ণয় কর :

$(3x+y{)}^3$

$=(3x{)}^3+3\times (3x{)}^2\times y+3\times \left(3x\right)\times (y{)}^2+(y{)}^3$

$=27x^3+3\times 9x^2\times xy+3\times 3x\times y^2+y^3$

$=27x^3+27x^{2}y+9x{y}^2+y^3$ (Ans.)

$(x^2+y{)}^3=(x^2{)}^3+3(x^2{)}^2\times y+3\times x^2\times (y{)}^2+(y{)}^3$

$=x^6+3\times x^4\times x\times y+3\times x^2\times x^2\times y^2+y^3$

$=x^6+3x^{4}y+3x^2{y}^2+y^3$ (Ans.)

$(5p+2q{)}^3$

$=(5p{)}^3+3\times (5p{)}^2\times 2q+3\times 5p\times (2q{)}^2+(2q{)}^3$

$=125p^3+3\times 25p^2\times 2q+3\times 5p\times 4q^2+8q^3$

$=125p^3+150p^{2}q+60p{q}^2+8q^3\mathbf{(}\mathit{A}\mathit{n}\mathit{s}\mathbf{.}\mathbf{)}$

$(a^{2}b+c^{2}d{)}^3$

$=(a^{2}b{)}^3+3\times (a^{2}b{)}^2\times \left(c^{2}d\right)+3\times \left(a^{2}b\right)\times (c^{2}d{)}^2+(c^{2}d{)}^3$

$=a^6{b}^3+3a^4{b}^2{c}^{2}d+3a^{2}b{c}^4{d}^2+c^6{d}^3$ (Ans.)

$(6p-7{)}^3$

$=(6p{)}^3-3\times (6p{)}^2\times 7+3\times \left(6p\right)\times (7{)}^2-(7{)}^3$

$=216p^3-3\times 36p^2\times 7+3\times 6p\times 49-343$

$=216p^3-756p^2+882p-343$ (Ans.)

$(ax-by{)}^3$

$=(ax{)}^3-3\times (ax{)}^2\times by+3\times ax\times (by{)}^2-(by{)}^3$

$=a^3{x}^3-3\times a^2{x}^2\times by+3\times ax\times b^2{y}^2-b^3{y}^3$

$=a^3{x}^3-3a^2{x}^2 by + 3ax{b}^2{y}^2-b^3{y}^3$ (Ans.)

${(2p^2 – 3r^2)}^3$

$=(2p^2{)}^3-3\times (2p^2{)}^2\times \left(3r^2\right)+3\times \left(2p^2\right)\times (3r^2{)}^2-(3r^2{)}^3$

$= 8p^6-3 \times 4p^4\times 3r^2 +3 \times 2p^2 \times 9r^4 - 27r^6$

$= 8p^6-36p^4-r^2 + 54p^2{r}^4 - 27r^6$ (Ans.)

$(x^3+2{)}^3=(x^3{)}^3+3.(x^3{)}^2.2+3.(x^3).(2{)}^2+(2{)}^3$

$=x^9+3.x^6.2+3.x^3.4+8$

$=x^9+6x^6+12x^3+8$ (Ans.)

$(2m+3n-5p{)}^3$

$=\left\{\right(2m+3n)-5p{\}}^3$

$=(2m+3n{)}^3-3.(2m+3n{)}^2.5p+3.(2m+3n).(5p{)}^2-(5p{)}^3$

$= \left\{(2m{)}^3+3.(2m{)}^2.3n+3.2m.(3n{)}^2+(3n{)}^3\right\} -3.\left\{\right(2m{)}^2+2.2m.3n+\left(3n{)}^2\right\}.5p+3.(2m+3n).25p^2-125p^3$

$=8m^3+3.4m^2.3n+3.2m.9n^2+27n^3-3\{4m^2+12mn+9n^2\}.5p+150m{p}^2+225p^{2}n-125p^3$

$=8m^3+36m^{2}n+54m{n}^2+27n^3-60m^{2}p-180mnp-135n^{2}p+150m{p}^2+225p^{2}n-125p^3$

$=8m^3+27n^3-125p^3+36m^{2}n-60m^{2}p+54m{n}^2+150m{p}^2-135n^{2}p+225p^{2}n-180mnp$ (Ans.)

$(x^2-y^2+z^2{)}^3=\{(x^2-y^2)+z^2{\}}^3$

$=(x^2-y^2{)}^3+3(x^2-y^2{)}^2{z}^2+3.(x^2-y^2).(z^2{)}^2+(z^2{)}^3$

$= (x^2{)}^3-3.(x^2{)}^2.y^2+3.x^2.(y^2{)}^2-(y^2{)}^3 +3z^2 (x^2{)}^2-2.x^2.y^2+(y^2{)}^2 +3z^4(x^2-y^2)+z^6$

$=x^6-3x^4{y}^2+3x^2{y}^4-y^6+3z^2(x^4-2x^2{y}^2+y^4)+3z^4{x}^2- 3y^{2}z+z^6$

$=x^6-3x^4{y}^2+3x^2{y}^4-y^6+3z^2{x}^4-6x^2{y}^2{z}^2+3y^4{z}^2+3z^4{x}^2- 3y^{2}z+z^6$

$=x^6-y^6+z^6-3x^4{y}^2+3x^2{y}^4+3x^4{z}^2+3y^4{z}^2+3x^2{z}^4- 3y^2{z}^4-6x^2{y}^2{z}^2$ (Ans.)

${(a^2{b}^2 – c^2{d}^2)}^2$

$= {\left(a^2{b}^2\right)}^3 - 3.{\left(a^2{b}^2\right)}^2.c^2{d}^2 + 3.a^2{b}^2.{\left(c^2{d}^2\right)}^2 - {\left(c^2{d}^2\right)}^3$

$= a^6{b}^6-3a^4{b}^4{c}^2{d}^2 + 3a^2{b}^2{c}^4{d}^4 - c^6{d}^6$ (Ans.)

${(a^{2}b – b^{3}c)}^3$

$= {\left(a^{2}b\right)}^3 - 3.{\left(a^{2}b\right)}^2-b^{3}c + 3.a^{2}b.{\left(b^{3}c\right)}^2 - {\left(bc\right)}^3$

$=a^{6}b3-3.a^4{b}^2.b^{3}c+3.a^{2}b.b^6{c}^2 - b^9{c}^3$

$= a^6{b}^3-3a^4{b}^{5}c + 3a^2{b}^7{c}^2 - b^9{c}^3$ (Ans.)

$(x^3-2y^3{)}^3$

$=(x^3{)}^3-3(x^3{)}^2.2y^3+3.x^3.(2y^3{)}^2-(2y^3{)}^3$

$=x^9-6x^6{y}^3+12x^3{y}^6-8y^9$ (Ans.)

$(11a-12b{)}^3$

$=(11a{)}^3-3.(11a{)}^2.\left(12b\right)+3.\left(11a\right).(12b{)}^2-(12b{)}^3$

$=1331a^3-3.121a^2.12b+3.11a.144b^2-1728b^3$

$=1331a^3-4356a^{2}b+4752a{b}^2-1728b^3$ (Ans.)

$(x^3+y^3{)}^3=(x^3{)}^3+3\left(x^3{)}^2\right(y^3)+3(x^3\left)\right(y^3{)}^2+(y^3{)}^3$

$=x^9+3x^6{y}^3+3x^3{y}^6+y^9$ (Ans.)