Let \({z}_{1}\) and \({z}_{2}\) be two complex numbers such that \({z}_{1}+{z}_{2}=5\) and \({z}_{1}^{3}+{z}_{2}^{3}=20+…

\(z_1+z_2=5\)

\(z_1^3+z_2^3=20+15i\)

\(\left|z_1^4+z_2^4\right|=?\)

\(z_1^3+z_2^3=(z_1+z_2)^3-3z_1z_2(z_1+z_2)\)

\(z_1^3+z_2^3=125-15z_1z_2\)

\(20+15i=125-15z_1z_2\)

\(\Rightarrow z_1z_2=7-i\)

Now,

\((z_1+z_2)^2=5^2\)

\(z_1^2+z_2^2+2z_1z_2=25\)

\(z_1^2+z_2^2=25-2(7-i)\)

\(=11+2i\)

\((z_1^2+z_2^2)^2=(11+2i)^2\)

\(z_1^4+z_2^4+2(7-i)^2=117+44i\)

\(z_1^4+z_2^4=21+72i\)

\(\left|z_1^4+z_2^4\right|=\sqrt{21^2+72^2}\)

\(=\sqrt{441+5184}\)

\(=\sqrt{5625}\)

\(=75\)