Question

Expand :

(i) $(2a+b{)}^3$

(ii) $(a-2b{)}^3$

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

(iv) $(x+5y{)}^3$

(v) ${(a+\frac{1}{a})}^3$

(vi) ${(2a-\frac{1}{2a})}^3$

Answer 1

(i) $(2a+b{)}^3=(2a{)}^3+(b{)}^3+3\times 2a\times b(2a+b\left)\right[(a+b{)}^3=a^3+b^3+3ab(a+b\left)\right](=8a^3+b^3+6ab(2a+b\left)\right)(=8a^2+b^3+12a^{2}b+6a{b}^2)$.

(ii) $(a-2b{)}^3=(a{)}^3-(2b{)}^3-3\times a\times 2b(a-2b\left)\right[(a-b{)}^3=a^3-b^3-3ab(a-b\left)\right]$

$=a^3-8b^3-6ab(a-2b)=a^3-8b^3-6a^{2}b+12a{b}^2$.

(iii) $(3x-2y{)}^3=(3x{)}^3-(2y{)}^3-3\times 3x\times 2y(2x-2y)=27x^3-8y^3-18xy(3x-2y)=27x^3-8y^3-54x^{2}y+36x{y}^2$.

(iv) $(x+5y{)}^3=(x{)}^3+(5y{)}^3+3\times x\times 5y(x+5y)=x^3+125y^3+15xy(x+5y)=x^3+125y^3+15x^{2}y+75x{y}^2$.

(v) ${(a+\frac{1}{a})}^3=a^3+{\left(\frac{1}{a}\right)}^3+3\times a\times \frac{1}{a}\times (a+\frac{1}{a})=a^3+\frac{1}{a^3}+3(a+\frac{1}{a})=a^3+\frac{1}{a^3}+3a+\frac{3}{a}$.

(vi) ${(2a-\frac{1}{2a})}^3=(2a{)}^3-{\left(\frac{1}{2a}\right)}^3-3\times 2a\times \frac{1}{2a}(2a-\frac{1}{2a})=8a^3-\frac{1}{8a^3}-3(2a-\frac{1}{2a})=8a^3-\frac{1}{8a^3}-6a+\frac{3}{2a}$.