Question:

We are given $z_1+z_2=5$ and $z_1^3+z_2^3=20+15i$. We need to find $|z_1^4+z_2^4|$. Step 1: Determine the value of $z_1z_2$. We use the identity $z_1^3+z_2^3=(z_1+z_2)^3-3z_1z_2(z_1+z_2)$. Substituting the given values: $$20+15i = (5)^3 - 3z_1z_2(5)$$ $$20+15i = 125 - 15z_1z_2$$ Rearranging the terms to solve for $z_1z_2$: $$15z_1z_2 = 125 - (20+15i)$$ $$15z_1z_2 = 105 - 15i$$ $$z_1z_2 = \frac{105-15i}{15}$$ $$z_1z_2 = 7-i$$ Step 2: Determine the value of $z_1^2+z_2^2$. We use the identity $z_1^2+z_2^2=(z_1+z_2)^2-2z_1z_2$. Substituting the known values: $$z_1^2+z_2^2 = (5)^2 - 2(7-i)$$ $$z_1^2+z_2^2 = 25 - 14 + 2i$$ $$z_1^2+z_2^2 = 11+2i$$ Step 3: Determine the value of $z_1^4+z_2^4$. We use the identity $z_1^4+z_2^4=(z_1^2+z_2^2)^2-2(z_1z_2)^2$. First, calculate $(z_1^2+z_2^2)^2$: $$(11+2i)^2 = 11^2 + 2(11)(2i) + (2i)^2$$ $$ = 121 + 44i - 4$$ $$ = 117+44i$$ Next, calculate $(z_1z_2)^2$: $$(7-i)^2 = 7^2 - 2(7)(i) + i^2$$ $$ = 49 - 14i - 1$$ $$ = 48-14i$$ Now, substitute these into the identity for $z_1^4+z_2^4$: $$z_1^4+z_2^4 = (117+44i) - 2(48-14i)$$ $$z_1^4+z_2^4 = 117+44i - 96+28i$$ $$z_1^4+z_2^4 = 21+72i$$ Step 4: Calculate $|z_1^4+z_2^4|$. We need to find the magnitude of $21+72i$: $$|21+72i| = \sqrt{21^2 + 72^2}$$ $$ = \sqrt{441 + 5184}$$ $$ = \sqrt{5625}$$ $$ = 75$$ Thus, $|z_1^4+z_2^4|=75$.