$\frac{a^3-b^3}{a^4-b^4}=\frac{(a-b)(a^2+ab+b^2)}{(a-b)(a^3+a^{2}b+a{b}^2+b^3)}=\frac{a^2+ab+b^2}{a^3+a^{2}b+a{b}^2+b^3}$

$\frac{a^3+b^3}{a^2-ab+b^2}=\frac{(a+b)(a^2-ab+b^2)}{a^2-ab+b^2}=a+b$

$\frac{a^5-b^5}{a^3-b^3}=\frac{(a-b)(a^4+a^{3}b+a^2{b}^2+a{b}^3+b^4)}{(a-b)(a^2+ab+b^2)}=\frac{a^4+a^{3}b+a^2{b}^2+a{b}^3+b^4}{a^2+ab+b^2}$

$\frac{a^5+b^5}{a^7+b^7}=\frac{(a+b)(a^4-a^{3}b+a^2{b}^2-a{b}^3+b^4)}{(a+b)(a^6-a^{5}b+a^4{b}^2-a^3{b}^3+a^2{b}^4-a{b}^5+b^6)}=\frac{a^4-a^{3}b+a^2{b}^2-a{b}^3+b^4}{a^6-a^{5}b+a^4{b}^2-a^3{b}^3+a^2{b}^4-a{b}^5+b^6}$

$\frac{a^3+a^{2}b+a{b}^2+b^3}{a^4-b^4}=\frac{a^3+a^{2}b+a{b}^2+b^3}{(a-b)(a^3+a^{2}b+a{b}^2+b^3)}=\frac{1}{a-b}$

$\frac{a^5+b^5}{a^4-a^{3}b+a^2{b}^2-a{b}^3+b^4}=\frac{(a+b)(a^4-a^{3}b+a^2{b}^2-a{b}^3+b^4)}{a^4-a^{3}b+a^2{b}^2-a{b}^3+b^4}=a+b$

$\frac{a^3-8}{a^4-16}=\frac{a^3-2^3}{a^4-2^4}=\frac{(a-2)(a^2+2a+4)}{(a-2)(a^3+a^{2}2+a{2}^2+2^3)}=\frac{(a-2)(a^2+2a+4)}{(a-2)(a^3+2a^2+4a+8)}=\frac{a^2+2a+4}{a^3+2a^2+4a+8}$

$\frac{a^3+27}{a^2-3a+9}=\frac{a^3+3^3}{a^2-3a+9}=\frac{(a+3)(a^2-3a+9)}{a62-3a+9}=a+3$

$\frac{a^5-32}{a^3-8}=\frac{a^5-2^5}{a^3-2^3}=\frac{(a-2)(a^4+a^{3}2+a^2{2}^2+a{2}^3+2^4)}{(a-2)(a^2+2a+4)}=\frac{a^4+2a^3+4a^2+8a+16}{a^2+2a+4}$

$\frac{a^5+32}{a^7+128}=\frac{a^5+2^5}{a^7+2^7}=\frac{(a+2)(a^4-2a^3+4a^2-8a+16)}{(a+2)(a^6-2a^5+4a^4-8a^3+16a^2-32a+64)}=\frac{a^4-2a^3+4a^2-8a+16}{a^6-2a^5+4a^4-8a^3+16a^2-32a+64}$

$\frac{a^3+2a^2+4a+8}{a^4-16}=\frac{a^3+2a^2+4a+8}{a^4-2^4}=\frac{a^3+2a62+4a+8}{(a-2)(a^3+2a62+4a+8)}=\frac{1}{a-2}$

$\frac{a^5+1}{a^4-a^3+a^2-a+1}=\frac{(a+1)(a^4-a^3+a^2-a+1)}{a^4-a^3+a^2-a+1}=a+1$