The given problem asks to simplify the following complex number into an imaginary number:

[ z_2 = \sqrt{-676} + \sqrt[3]{-1331} - \sqrt[5]{(-7)^5} + \sqrt[3]{(-64)^2} ]

Let's break this down step by step:

1. First term: $\sqrt{-676}$ $\sqrt{-676} = \sqrt{676} \cdot \sqrt{-1} = 26i$ where $i$ is the imaginary unit.

2. Second term: $\sqrt[3]{-1331}$ [ \sqrt[3]{-1331} = -\sqrt[3]{1331} = -11 ]

3. Third term: $\sqrt[5]{(-7)^5}$ $\sqrt[5]{(-7)^5} = -7$

4. Fourth term: $\sqrt[3]{(-64)^2}$ [ (-64)^2 = 4096 \quad \Rightarrow \quad \sqrt[3]{4096} = \sqrt[3]{16^3} = 16 ]

Now, summing these terms together:

$z_2 = 26i + (-11) + (-7) + 16$ Simplifying the real terms: $z_2 = 26i + (-11 - 7 + 16) = 26i - 2$

Thus, the simplified form of the complex number is: $z_2 = 26i - 2$

To make it purely imaginary, we would disregard the real part $-2$, and the imaginary part is $26i$.

Would you like further details or have any questions?

Related Questions:

1. How do you simplify other types of complex numbers?
2. What is the significance of the imaginary unit $i$?
3. How do cube roots of negative numbers work in complex numbers?
4. Can you explain higher-order roots, such as fifth roots, in more detail?
5. What are the properties of imaginary and complex numbers in general?

Tip:

Always check the roots carefully, especially for negative radicands, as they often involve the imaginary unit $i$.