worksheet 1.3

1. $(a+b{)}^4$
   $=a^4+4a^{3}b+6a^2{b}^2+4a{b}^3+b^4$

2. $(x+y{)}^6$
   $=a^6+6x^{5}y+15x^4{y}^2+20x^3{y}^3+15x^2{y}^4+6x{y}^5+y^6$

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

4. $(x-y{)}^5$
   $=x^5+5x^4(-y)+10x^3(-y{)}^2+10x^3(-y{)}^3+5x(-y{)}^4+(-y{)}^5$
   $=x^5-5x^{4}y+10x^3{y}^2-10x^3{y}^3+5x{y}^4-y^5$

5. $(a+3b{)}^3$
   $=a^3+3a^2(2b)+3a(3b{)}^2+(3b{)}^3$
   $=a^3+6a^{2}b+9a{b}^2+27b^3$

6. $(2x-y{)}^4$
   $=(2x{)}^4+4(2x{)}^3(-y)+6(2x{)}^2(-y{)}^2+4(2x)(-y{)}^3+(-y{)}^4$
   $=16x^4-32x^{3}y+24x^2{y}^2-8x{y}^3+y^4$

7. $(3a-2b{)}^3$
   $=(3a{)}^3+3(3a{)}^2(-2b)+3(3a)(-2b{)}^2+(-2b{)}^3$
   $=3a^3-54a^{2}b+36a{b}^2-8b^3$

8. $(\frac{1}{x}+x{)}^4$
   $=(\frac{1}{x}{)}^4+4(\frac{1}{x}{)}^3(x)+6(\frac{1}{x}{)}^2(x{)}^2+4(\frac{1}{x})(x{)}^3+x^4$
   $=\frac{1}{x^4}+\frac{4}{x^2}+6+4x^2+x^4$

9. $(2^x+2{)}^4$
   $=(2^x{)}^4+4(2^x{)}^3(2)+6(2^x{)}^2(2{)}^2+4(2^x)(2{)}^3+2^4$
   $=2^{4x}+8(2^{3x})+24(2^{2x})+32(2^x)+16$

10. $(2x-\frac{1}{x}{)}^3$
    $=(2x{)}^3+3(2x{)}^2(\frac{-1}{x})+3(2x)(\frac{-1}{x}{)}^2+(\frac{-1}{x}{)}^3$
    $=8x^3-12x+\frac{6}{x}-\frac{1}{x^3}$

11. $(3-3^x{)}^5$
    $=3^5+5(3{)}^4(-3^x)+10(3{)}^3(-3^x{)}^2+10(3{)}^2(-3^x{)}^3+5(3)(-3^x{)}^4+(-3^x{)}^5$
    $=243-405(3^x)+270(3^{2x})-90(3^{3x})+15(3^{4x})-3^{5x}$

12. $(\frac{2}{x}-x{)}^5$
    $=(\frac{2}{x}{)}^5+5(\frac{2}{x}{)}^4(-x)+10(\frac{2}{x}{)}^3(-x{)}^2+10(\frac{2}{x}{)}^2(-x{)}^3+5(\frac{2}{x})(-x{)}^4+(-x{)}^5$
    $=\frac{32}{x^5}+\frac{-80x}{x}+\frac{80x^2}{x}+\frac{-40x^3}{x}+\frac{10x^4}{x}-x^5$
    $=\frac{32}{x^5}-80+80x-40x^2+10x^3-x^5$

13. $(x+y{)}^5+(2x-y{)}^5$
    $=x^5+5x^{4}y+10x^3{y}^2+10x^2{y}^3+5x{y}^4+y^5+(2x{)}^5+5(2x{)}^4(-y)+10(2x{)}^3(-y{)}^2+10(2x{)}^2(-y{)}^3+5(2x)(-y{)}^4+(-y{)}^5)$
    $=33x^5-75x^{4}y+80x^3{y}^2-30x^2{y}^3+15x{y}^4$

14. $(x-y{)}^7+(x+y{)}^7$
    $=x^7-7x^{6}y+21x^5{y}^2-35x^4{y}^3+35x^3{y}^4-21x^2{y}^5+7x{y}^6-y^7+7x^{6}y+21x5y^2-35x^4{y}^3+35x^3{y}^4-21x^2{y}^5+7x{y}^6-y^7$
    $=2x^7+42x^5{y}^2+70x^3{y}^4+14x{y}^6$