Suddenly in the mood for algebra

$x+y=1$

$(x+y)^2=1^2$

$x^2+y^2+2xy=1$

$2+2xy=1$

$2xy=-1$

$xy=-\frac{1}{2}$

Therefore, $x^3+y^3=(x+y)(x^2+y^2-xy)=1(2-(-\frac{1}{2}))=2+\frac{1}{2}=\frac{2\times2+1}{2}=\frac{4+1}{2}=\frac{5}{2}$ .

Therefore, $a\times b=5\times2=\boxed{10}$

$x+y=1$

$(x+y)^2=1\Rightarrow x^2+y^2+2xy=1$

$x^2+y^2=2$

$2+2xy=1\Rightarrow xy=-\dfrac{1}{2}$

$(x+y)^3=1\Rightarrow x^3+y^3+3x^2y+3xy^2=1$

$x^3+y^3+3xy(x+y)=1\Rightarrow x^3+y^3+3 \times-\dfrac{1}{2}\times1=1\Rightarrow x^3+y^3=\dfrac{5}{2}$

$a\times b=5\times2=\boxed{10}$