a, \(a+b=10\Rightarrow\left(a+b\right)^2=10^2\Rightarrow a^2+2ab+b^2=100\)

\(\Rightarrow a^2+b^2=100-2ab\Rightarrow a^2+b^2=100-2.4\Rightarrow a^2+b^2=100-8\)

\(\Rightarrow a^2+b^2=92\). Vậy \(a^2+b^2=92\)

b, \(a+b=10\Rightarrow\left(a+b\right)^3=10^3\Rightarrow a^3+3a^2b+3ab^2+b^3=1000\)

\(\Rightarrow a^3+b^3+3ab\left(a+b\right)=1000\Rightarrow a^3+b^3+3.4.10=1000\)

\(\Rightarrow a^3+b^3+120=1000\Rightarrow a^3+b^3=880\). Vậy \(a^3+b^3=880\)

c, \(a+b=10\Rightarrow\left(a+b\right)^4=10000\)

\(\Rightarrow a^4+4a^3b+6a^2b^2+4ab^3+b^4=10000\)

\(\Rightarrow a^4+b^4+4ab\left(a^2+b^2\right)+6\left(ab\right)^2=10000\)

\(\Rightarrow a^4+b^4+4.4.92+6.4^2=10000\Rightarrow a^4+b^4+992+96=10000\)

\(\Rightarrow a^4+b^4=8912\). Vậy \(a^4+b^4=8912\)

d, \(a+b=10\Rightarrow\left(a+b\right)^5=100000\)

\(\Rightarrow a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5=100000\)

\(\Rightarrow a^5+b^5+5ab\left(a^3+b^3\right)+10a^2b^2\left(a+b\right)=100000\)

\(\Rightarrow a^5+b^5+5.4.880+10.4^2.10=100000\)

\(\Rightarrow a^5+b^5+17600+1600=100000\Rightarrow a^5+b^5=80800\)

Vậy \(a^5+b^5=80800\)