# Use binomial theorem to find the complex number

• chwala
In summary, the equation (a+bi)^5 can be expanded into a simplified form of a^5+5a^4bi-10a^3b^2-10a^2b^3i+5ab^4+b^5i. This can also be expressed as the real part (a^5-10a^3b^2+5ab^4) and the imaginary part (5a^4b-10a^2b^3+b^5).
chwala
Gold Member
Homework Statement
If ##z=a+bi##, where ##a## and ##b## are real, use binomial theorem to find the real and imaginary parts of ##z^5##
Relevant Equations
Complex numbers
This is also pretty easy,
##z^5=(a+bi)^5##
##(a+bi)^5= a^5+\dfrac {5a^4bi}{1!}+\dfrac {20a^3(bi)^2}{2!}+\dfrac {60a^2(bi)^3}{3!}+\dfrac {120a(bi)^4}{4!}+\dfrac {120(bi)^5}{5!}##
##(a+bi)^5=a^5+5a^4bi-10a^3b^2-10a^2b^3i+5ab^4+b^5i##
##\bigl(\Re (z))=a^5-10a^3b^2+5ab^4##
##\bigl(\Im (z))= 5a^4b-10a^2b^3+b^5##

Any other variation, combinations may also work...

Last edited:
chwala said:
##(a+bi)^5= a^5+\dfrac {5a^4bi}{1!}-\dfrac {20a^3(bi)^2}{2!}-\dfrac {60a^2(bi)^3}{3!}+\dfrac {120a(bi)^4}{4!}+\dfrac {120(bi)^5}{5!}##
You should have ##+## throughout that equation. That said, you got the right answer.

PeroK said:
You should have ##+## throughout that equation. That said, you got the right answer.
True, let me amend that...

## 1. What is the binomial theorem?

The binomial theorem is a mathematical formula that allows us to expand expressions of the form (a + b)^n, where a and b are constants and n is a positive integer.

## 2. How do you use the binomial theorem to find a complex number?

To use the binomial theorem to find a complex number, we first need to express the complex number in the form (a + b)^n. Then, we can use the binomial theorem to expand this expression and simplify it to find the complex number.

## 3. Can the binomial theorem be used for negative exponents?

Yes, the binomial theorem can be used for negative exponents. In this case, we use the formula (a + b)^n = 1/(a + b)^(-n) to expand the expression and find the complex number.

## 4. What are some real-life applications of the binomial theorem?

The binomial theorem has many real-life applications, such as in probability and statistics, finance, and engineering. It is also used in fields like physics and chemistry to model and solve complex problems.

## 5. Are there any limitations to using the binomial theorem to find complex numbers?

One limitation of using the binomial theorem to find complex numbers is that it can only be used for expressions in the form (a + b)^n. It cannot be applied to more complex expressions or those with variables in the exponent. Additionally, it may not always yield exact solutions, and some rounding or approximation may be necessary.

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