hadroneater said:Homework Statement
Find the modulus for:
(3 - 4i)^10/(2 - i)^8
Homework Equations
The Attempt at a Solution
I tried putting the two terms in their exponential forms. Then do simple exponential operations and convert back to cartesian form to get the modulus.
3 - 4i = [itex]5e^{arctan(-4/3)}[/itex]
2 - i = [itex]\sqrt{5}e^{arctan(1/2)}[/itex]
However the arguments aren't going to be exact numbers. Is there another way to approach this question?
The modulus of a complex number is its distance from the origin on the complex plane. In this case, the modulus of (3-4i)^10/(2-i)^8 can be calculated by taking the absolute value of the complex number. This can be done by finding the square root of the sum of the squares of the real and imaginary parts. The final answer will be a positive real number.
To simplify this expression, we can use the properties of exponents and the quotient of powers rule. First, we can expand (3-4i)^10 and (2-i)^8 using the binomial theorem. Then, we can divide the two expressions by writing them with a common base and subtracting the exponents. Finally, we can simplify the resulting expression by combining like terms and simplifying any remaining complex numbers.
No, the modulus of a complex number is always a positive real number. This is because it represents the distance from the origin on the complex plane, which can never be negative.
When a complex number is raised to a power, its modulus is also raised to that power. This means that the distance from the origin on the complex plane will increase or decrease depending on the value of the exponent.
The modulus of a complex number is important in operations involving complex numbers because it helps us understand the magnitude or size of the number. It also allows us to find the distance between two complex numbers and to compare the sizes of different complex numbers. In addition, the modulus is used in polar form to represent complex numbers, making it a crucial concept in complex number operations.