QuestionQuestion
Simplify the expression to a + bi form: $(6+i)^{2}$

Studdy Solution

STEP 1

What is this asking? We need to multiply the complex number $6+i$ by itself and then write the answer in the standard form for complex numbers, which is $a + bi$ . Watch out! Remember that $i^2$ is equal to $-1$ , not $1$ !
Also, don't forget to combine like terms after multiplying.

STEP 2

1. Expand the expression.
2. Simplify the expression.

STEP 3

Alright, let's expand this expression like we're unfolding a treasure map!
We have $(6+i)^2$ , which is the same as $(6+i) \cdot (6+i)$ .
We'll use the FOIL method (First, Outer, Inner, Last) to multiply these two binomials.

STEP 4

First: $6 \cdot 6 = 36$ . Outer: $6 \cdot i = 6i$ . Inner: $i \cdot 6 = 6i$ . Last: $i \cdot i = i^2$ .

STEP 5

Now, let's put it all together: $36 + 6i + 6i + i^2$ .

STEP 6

Time to simplify!
Remember that magical property of $i$ : $i^2 = -1$ .
Let's substitute that in: $36 + 6i + 6i + (-1)$ .

STEP 7

Now, combine those like terms.
The real parts are $36$ and $-1$ , and the imaginary parts are $6i$ and $6i$ .

STEP 8

Adding the real parts, we get $36 + (-1) = 35$ .
Adding the imaginary parts, we get $6i + 6i = 12i$ .

STEP 9

So, our final simplified expression is $35 + 12i$ .
Boom!

STEP 10

The simplified form of $(6+i)^2$ is $35 + 12i$ .

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