[tex]\displaystyle \sf \left(\frac{30}{x\cdot \sqrt[3]{\sf 35-x^3}}\right) =x+\sqrt[3]{\sf 35-x^3} \\\\\\ \text{fatora\c c\~ao para o cubo da soma} : \\\\ (a+b)^3=a^3+b^3+3 ab\cdot (a+b) \\\\\ \text{Elevando ao cubo ambos os lados da express\~ao}\\\\ \left(\frac{30}{x\cdot \sqrt[3]{\sf 35-x^3}}\right)^3 =\left(x+\sqrt[3]{\sf 35-x^3}\right)^3[/tex]

[tex]\displaystyle \sf \frac{30^3}{x^3\cdot (35-x^3)} =x^3+\left(\sqrt[3]{\sf 35-x^3}\right)^3+3\left(x\cdot \sqrt[3]{\sf 35-x^3}\right)\cdot \underbrace{\sf \left(x+\sqrt[3]{\sf 35-x^3}\right)}_{\displaystyle \sf \left(\frac{30}{x\cdot \sqrt[3]{\sf 35-x^3}}\right)} \\\\\\ \frac{30^3}{x^3\cdot (35-x^3)}=x^3+35-x^3+3\left(x\cdot \sqrt[3]{\sf 35-x^3}\right)\cdot \frac{30}{\left(x\cdot \sqrt[3]{\sf 35-x^3}\right)}[/tex]

[tex]\displaystyle \sf \frac{30^3}{x^3\cdot (35-x^3)}=35+3\cdot 30 \\\\\\ \frac{30^3}{35x^3-x^6}=35+90=125\\\\\\ \frac{3^3.10^3}{125}=(35x^3-x^6)\\\\\\ \frac{27.1000}{125}=35x^3-x^6\\\\\\ 27\cdot 8=35x^3-x^6 \\\\ x^6-35x^3+216=0 \\\\ \text{Fa\c ca} :\\\\ x^3=m \to x^6 = m^2.\\\\ Assim, \\\\ m^2-35m+216 = 0[/tex]

[tex]\displaystyle \sf m=\frac{-(-35)\pm\sqrt{(-35)^2-4.216.1}}{2.1}\\\\\\ m=\frac{35\pm\sqrt{1225-864}}{2} \\\\\ m =\frac{35\pm \sqrt{361}}{2} \\\\\\ m = \frac{35\pm19}{2}\\\\\\ m= \frac{35+19}{2}=\frac{54}{2}\to m =27\\\\\\ m=\frac{35-19}{2}=\frac{16}{2}\to m=8[/tex]

Então :

[tex]\displaystyle \sf m=27\to x^3=27\to x= 3=a\\\\ m=8 \to x^3=8\to x =2=b \\\\ \tex{Logo : }\\\\ a+b = 3+2\\\\ \large\boxed{\sf \ a+b = 5\ }\checkmark[/tex]