Key Concepts and Formulas

• Sum of Cubes Identity: $a^3 + b^3 = (a+b)^3 - 3ab(a+b)$
• Sum of Squares Identity: $a^2 + b^2 = (a+b)^2 - 2ab$
• Sum of Fourth Powers Identity: $a^4 + b^4 = (a^2+b^2)^2 - 2(ab)^2$
• Modulus of a Complex Number: For a complex number $Z = x+iy$, its modulus is $|Z| = \sqrt{x^2+y^2}$.

Step-by-Step Solution

Step 1: Calculate the product $z_1 z_2$

Why this step? To find higher powers like $z_1^4+z_2^4$, we first need to determine the product $z_1 z_2$. This is because the identities for sums of powers often involve both the sum ($z_1+z_2$) and the product ($z_1 z_2$). We can find $z_1 z_2$ using the given sum of cubes.

We apply the sum of cubes identity $a^3 + b^3 = (a+b)^3 - 3ab(a+b)$, by substituting $a=z_1$ and $b=z_2$: $z_1^3 + z_2^3 = (z_1+z_2)^3 - 3z_1 z_2(z_1+z_2)$

Now, substitute the given values $z_1+z_2 = 5$ and $z_1^3 + z_2^3 = 20 + 15i$ into this equation: $20 + 15i = (5)^3 - 3z_1 z_2 (5)$ $20 + 15i = 125 - 15z_1 z_2$

To isolate $15z_1 z_2$, we rearrange the equation: $15z_1 z_2 = 125 - (20 + 15i)$ $15z_1 z_2 = 125 - 20 - 15i$ $15z_1 z_2 = 105 - 15i$

Finally, divide by 15 to find $z_1 z_2$: $z_1 z_2 = \frac{105 - 15i}{15}$ $z_1 z_2 = 7 - i$

Step 2: Calculate the sum of squares $z_1^2 + z_2^2$

Why this step? The sum of fourth powers identity, $a^4+b^4 = (a^2+b^2)^2 - 2(ab)^2$, requires us to know $z_1^2+z_2^2$ and $(z_1z_2)^2$. We already found $z_1z_2$, so the next logical step is to find $z_1^2+z_2^2$.

We use the sum of squares identity $a^2 + b^2 = (a+b)^2 - 2ab$: $z_1^2 + z_2^2 = (z_1+z_2)^2 - 2z_1 z_2$

Substitute the known values $z_1+z_2 = 5$ and $z_1 z_2 = 7-i$: $z_1^2 + z_2^2 = (5)^2 - 2(7-i)$ $z_1^2 + z_2^2 = 25 - (14 - 2i)$ $z_1^2 + z_2^2 = 25 - 14 + 2i$ $z_1^2 + z_2^2 = 11 + 2i$

Step 3: Calculate the sum of fourth powers $z_1^4 + z_2^4$

Why this step? This is the penultimate step before finding the modulus. We now have all the components needed to apply the sum of fourth powers identity.

We use the identity $a^4 + b^4 = (a^2+b^2)^2 - 2(ab)^2$. Substitute $a=z_1$ and $b=z_2$: $z_1^4 + z_2^4 = (z_1^2+z_2^2)^2 - 2(z_1 z_2)^2$

Substitute $z_1^2 + z_2^2 = 11+2i$ and $z_1 z_2 = 7-i$: $z_1^4 + z_2^4 = (11+2i)^2 - 2(7-i)^2$

Expanding $(11+2i)^2$: $(11+2i)^2 = 11^2 + 2(11)(2i) + (2i)^2 = 121 + 44i - 4 = 117 + 44i$

Expanding $(7-i)^2$: $(7-i)^2 = 7^2 - 2(7)(i) + i^2 = 49 - 14i - 1 = 48 - 14i$

Substitute these results back into the equation for $z_1^4 + z_2^4$: $z_1^4 + z_2^4 = (117+44i) - 2(48-14i)$ $z_1^4 + z_2^4 = 117+44i - 96+28i$

Combine the real and imaginary parts: $z_1^4 + z_2^4 = (117-96) + (44+28)i$ $z_1^4 + z_2^4 = 21 + 72i$

Step 4: Calculate the modulus $\left|z_1^4 + z_2^4\right|$

Why this step? The final requirement of the problem is to find the modulus of the complex number $z_1^4+z_2^4$.

We have $z_1^4 + z_2^4 = 21 + 72i$. Using the definition of the modulus for a complex number $x+iy$, which is $\sqrt{x^2+y^2}$: $\left|z_1^4 + z_2^4\right| = \left|21 + 72i\right| = \sqrt{21^2 + 72^2}$

Calculate the squares: $21^2 = 441$ $72^2 = 5184$

Now, sum them and take the square root: $\left|z_1^4 + z_2^4\right| = \sqrt{441 + 5184}$ $\left|z_1^4 + z_2^4\right| = \sqrt{5625}$ $\left|z_1^4 + z_2^4\right| = 75$

Common Mistakes & Tips

• Incorrectly expanding $(a+bi)^2$ or $(a-bi)^2$. Remember $(a+bi)^2 = a^2 - b^2 + 2abi$.
• Errors in sign while distributing negative signs or combining terms.
• Miscalculating squares of numbers.

Summary

This problem effectively demonstrates how to systematically evaluate expressions involving higher powers of complex numbers by leveraging basic algebraic identities. By iteratively finding the product ($z_1z_2$) and sums of increasing powers ($z_1^2+z_2^2$, then $z_1^4+z_2^4$), we can break down a complex problem into manageable steps. The final step involves calculating the modulus of the resulting complex number. The problem highlights the importance of precise algebraic manipulation and complex number arithmetic.

The final answer is 75, which corresponds to option (D).

Final Answer The final answer is $\boxed{75}$, which corresponds to option (D).