Let z 1 and z 2 be two complex number such that z 1 + z 2 = 5 and z 1 3 + z 2 3 = 20 + 15 i . Then z 1 4 + z…

Given: $z_1+z_2=5 \& {z_1}^3+{z_2}^3=20+15i$

Now, solving ${z_1}^3+{z_2}^3=20+15i$

$\Rightarrow {\left(z_1+z_2\right)}^3-3z_1{z}_2\left(z_1+z_2\right)=20+15i$

$\Rightarrow {\left(5\right)}^3-3z_1{z}_2\left(5\right)=20+15i$

$\Rightarrow 125-15z_1{z}_2=20+15i$

$\Rightarrow z_1{z}_2=7-i$

Now, finding ${z_1}^4+{z_2}^4={\left({z_1}^2+{z_2}^2\right)}^2-2{\left(z_1{z}_2\right)}^2$

$\Rightarrow {z_1}^4+{z_2}^4={\left({\left(z_1+z_2\right)}^2-2z_1{z}_2\right)}^2-2{\left(z_1{z}_2\right)}^2$

$\Rightarrow {z_1}^4+{z_2}^4={\left(25-2\left(7-i\right)\right)}^2-2{\left(7-i\right)}^2$

$\Rightarrow {z_1}^4+{z_2}^4={\left(11+2i\right)}^2-2\left(48-14i\right)$

$\Rightarrow {z_1}^4+{z_2}^4=\left(121-4+44i\right)-96+28i$

$\Rightarrow {z_1}^4+{z_2}^4=21+72i$

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

$\Rightarrow \left|{z_1}^4+{z_2}^4\right|=\sqrt{5625}$

$\Rightarrow \left|{z_1}^4+{z_2}^4\right|=75$