Finding the Value of $(1+i)^4$

This section provides a detailed step-by-step explanation to find the value of the complex number expression $(1+i)^4$. We will simplify the expression using basic algebraic rules and the properties of the imaginary unit, $i$.

Understanding Powers of Complex Numbers

A complex number is typically written in the form $a + bi$, where '$a$' is the real part, '$b$' is the imaginary part, and '$i$' is the imaginary unit. The imaginary unit $i$ is defined such that $i^2 = -1$. To find the value of $(1+i)^4$, we need to multiply $(1+i)$ by itself four times.

Method: Step-by-Step Simplification

A common strategy for calculating higher powers of complex numbers is to first compute the square of the base number and then square the result. This simplifies the process.

Step 1: Calculate the square, $(1+i)^2$

We can expand $(1+i)^2$ using the binomial expansion formula $(a+b)^2 = a^2 + 2ab + b^2$.

Here, $a = 1$ and $b = i$. Applying the formula:

Recall that $i^2 = -1$. Substitute this value into the equation:

Now, simplify the expression by combining the real terms:

So, we found that $(1+i)^2$ simplifies to $2i$.

Step 2: Calculate $(1+i)^4$ using the result from Step 1

We can rewrite $(1+i)^4$ as $((1+i)^2)^2$. This allows us to use the result we just calculated.

Substitute the value $(1+i)^2 = 2i$ into the equation:

Now, we need to square the complex number $2i$. This means squaring both the coefficient (2) and the imaginary unit ($i$):

Calculate the squares:

Again, substitute $i^2 = -1$:

Performing the final multiplication:

Therefore, the value of $(1+i)^4$ is $-4$.

Summary of the Calculation

The process to find the value of $(1+i)^4$ involved two main steps:

• First, we computed $(1+i)^2$, which resulted in $2i$.
• Second, we squared this result $(2i)^2$, which yielded $-4$.

This step-by-step approach simplifies the calculation of powers for complex numbers.