https://file.aiquickdraw.com/mathbot/file/17266451356745bnsxkh5.jpg

The problem asks to find and plot the roots of the following complex numbers on the complex plane:

1. \( \sqrt[5]{1} \) (fifth root of 1)
2. \( \sqrt{-4} \) (square root of -4)
3. \( \sqrt{-24 - 10j} \) (square root of \( -24 - 10j \))
4. \( \sqrt[3]{1 - j} \) (cube root of \( 1 - j \))
5. \( \frac{7 - j}{\sqrt{-3 + 4j}} \) (a complex fraction)

### Solution Steps:

1. **Fifth Root of 1**: 
   \( \sqrt[5]{1} \) gives 5 roots in total on the unit circle in the complex plane. They are given by \( e^{2\pi ik / 5} \) for \( k = 0, 1, 2, 3, 4 \).

2. **Square Root of -4**: 
   \( \sqrt{-4} \) is \( 2j \), because \( j^2 = -1 \), where \( j \) is the imaginary unit.

3. **Square Root of \( -24 - 10j \)**:
   Convert \( -24 - 10j \) to polar form, and use the formula for the square root in polar coordinates.

4. **Cube Root of \( 1 - j \)**:
   Convert \( 1 - j \) into polar form, then find its cube roots by dividing the argument by 3.

5. **Complex Fraction** \( \frac{7 - j}{\sqrt{-3 + 4j}} \):
   First, calculate the square root of \( -3 + 4j \) and then simplify the fraction.

Would you like a detailed explanation for each step, or just the final answers for these roots?

### Related Questions:
1. How do you convert a complex number into polar form?
2. What is De Moivre's theorem and how does it apply to roots of unity?
3. How do you find the cube roots of a complex number?
4. What is the geometric interpretation of complex roots on the complex plane?
5. How do you simplify expressions with complex fractions?

#### Tip:
When plotting complex numbers, their modulus represents the distance from the origin, and their argument represents the angle from the positive real axis.

Найти и изобразить на комплексной плоскости все значения следующих корней: 1) 5√1 ; 2) √-4 ; 3) √-24 - 10j ; 4) 3√1 - j ; 5) (7 - j) / √(-3 + 4j)

This math problem covers finding the roots of several complex numbers, including the fifth root of 1, the square root of -4, and other complex expressions. The solution involves applying concepts such as De Moivre's Theorem and working with polar coordinates. Suitable for advanced high school students.

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