1.(x+y)^4+x^4+y^4=2(x^2+xy+y^2)^22.(x-2y)x^3-(y-2x)y^3=(x+y)(x-y)^33.(3^n+1)+(5^n+2)-(3^n)-(5^n)=24(5^n+(3^n-1))4.(a^5n)+(a^n)+1=((a^3n)-(a^2n)+1)((a^2n)+(a^n)+1).(3^n+2)+(5^n+2)-(3^n)-(5^n)=24(5^n+(3^n-1))

Question

1.(x+y)^4+x^4+y^4=2(x^2+xy+y^2)^2
2.(x-2y)x^3-(y-2x)y^3=(x+y)(x-y)^3
3.(3^n+1) + (5^n+2) -(3^n)-(5^n)=24(5^n+(3^n-1))
4.(a^5n)+(a^n)+1=((a^3n)-(a^2n)+1)((a^2n)+(a^n)+1)
.(3^n+2) + (5^n+2) -(3^n)-(5^n)=24(5^n+(3^n-1))

Answer 1

1、
x^4+y^4+(x+y)^4
=x^4+2x^2y^2+y^4+(x+y)^4-2x^2y^2
=(x^2+y^2)^2-2x^2y^2+(x+y)^4
=[(x+y)^2-2xy]^2-2x^2y^2+(x+y)^4
=[(x+y)^2]^2-4xy(x+y)^2+4x^2y^2-2x^2y^2+(x+y)^4
=2(x+y)^4-4xy(x+y)^2+2x^2y^2
=2[(x+y)^4-2xy(x+y)^2+(xy)^2]
=2[(x+y)^2-xy]^2
=2(x^2+xy+y^2)^2
2、
(x-2y)x^3-(y-2x)y^3
=x^4-y^4+2xy^3-2x^3y
=(x^2+y^2)(x^2-y^2)-2xy(x^2-y^2)
=(x^2-2xy+y^2)(x^2-y^2)
=(x-y)^2(x+y)
3、
3^(n+2)+5^(n+2)-3^n-5^n
=3^(n-1)*(3^3-3)+5^n(5^2-1)
=24*3^(n-1)+24*5^n
=24*[3^(n-1)+5^n]
4、
a^5n+a^n+1
=a^5n-a^2n+a^2n+a^n+1
=a^2n(a^3n-1)+(a^2n+a^n+1)
=a^2n*(a^n-1)(a^2n+a^n+1)+(a^2n+a^n+1)
=(a^3n-a^2n+1)(a^2n+a^n+1)