15^x : 5^y = 45^y

=> 15^x = 225^y

=> 15^x = 15^2y

=> x = 2y

=> \(\frac{x}{y}=\frac{2}{1}\)

Ta có: \(\left(x+y\right)^5=x^5+y^5\)

\(\Leftrightarrow\left(x+y\right)^5-x^5-y^5=0\)

\(\Leftrightarrow x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5-x^5-y^5=0\)

\(\Leftrightarrow5x^4y+10x^3y^2+10x^2y^3+5xy^4=0\)

\(\Leftrightarrow\left(5x^4y+5xy^4\right)+\left(10x^3y^2+10x^2y^3\right)=0\)

\(\Leftrightarrow5xy\left(x^3+y^3\right)+10x^2y^2\left(x+y\right)=0\)

\(\Leftrightarrow5xy\left(x+y\right)\left(x^2-xy+y^2\right)+10x^2y^2\left(x+y\right)=0\)

\(\Leftrightarrow5xy\left(x+y\right)\left(x^2+xy+y^2\right)=0\)

\(\Rightarrow\)hoặc 5xy = 0 hoặc x + y = 0 hoặc \(x^2+xy+y^2=0\)

\(+)5xy=0\Rightarrow\orbr{\begin{cases}x=0\\y=0\end{cases}}\)

\(+)x+y=0\Rightarrow x=-y\)(hai số đối)

\(+)x^2+xy+y^2=0\)

\(\Leftrightarrow x^2+2.x.\frac{y}{2}+\frac{y^2}{4}+\frac{3y^2}{4}=0\)

\(\Leftrightarrow\left(x+\frac{y}{2}\right)^2+\frac{3y^2}{4}=0\)

Mà \(\left(x+\frac{y}{2}\right)^2+\frac{3y^2}{4}\ge0\)

(Dấu "="\(\Leftrightarrow x=y=0\))

Vậy x và y là hai số đối

(x+y)⁵-x⁵-y⁵=0

=>x⁵+y⁵+5x⁴y+10x³y²+10x²y³+5xy⁴-x⁵=0

=>5x⁴y+10x³y²+10x²y³+5xy⁴=0

=>5xy(x³+y³+2x²y+2xy²)=0

=>x³+y²+2x²y+2xy²=0

=>(x+y)(x²-xy+y²)+2xy(x+y)=0

=>(x+y)(x²-xy+y²+2xy)=0

=>(x+y)(x²+xy+y²)=0

=>x+y=0 hoặc x²+y²+xy=0

Vậy x+y=0(đpcm)

(x+y)⁵ =x⁵+y⁵ = (x+y)(x⁴ +....+y⁴)

=>(x+y) [(x+y)⁴-(x⁴+...+y⁴)] =0 vì [....] >0

=> x+y =0