Question

How do you simplify $\left(-8+9i\right)\left(6+9i\right)$?

Answer 1

We are given to solve an expression given in imaginary numbers. For this we will multiply each term with each other term. And we use some more properties of iota. While multiplying, we take care of the sign of the term obtained.

Complete step by step solution:
We are given $\left(-8+9i\right)\left(6+9i\right)$. We use simple rules of multiplication where we first take the first term of each bracket and find the product. After that we take the second term of the second bracket and multiply it with the first term of the first bracket. We do this till each term of the two brackets gets multiplied with each other term.
Consider:
$\left(-8+9i\right)\left(6+9i\right)$
The first term obtained of the resultant expression will be:
$-8\times 6=-48$
The second term obtained will be:
$-8\times 9i=-72i$
The third term obtained after multiplying the terms will be:
$9i\times 6=54i$
And the last term obtained after multiplying the last two terms of each bracket will be:
$9i\times 9i=81 \times i^2$
Now, we use the fact that:
$i^2=-1$
So, we get:
$9i\times 9i=81 \times -1=-81$
After combining the terms obtained we simply put them together and we get:
$\left(-8+9i\right)\left(6+9i\right)=-48-72i+54i-81$
We then obtain:
$\left(-8+9i\right)\left(6+9i\right)=-129-18i$
Hence, we have solved the expression.

Note:
We can also use the following formula while solving:
$\left(a+bi\right)\left(c+di\right)=ac-bd+\left(ad+bc\right)i$
We can put the values of $a$ and $b$ directly and obtain the result. Also you should be aware while doing the calculations because such multiplication can lead to calculation mistakes. Moreover, using the formula there is a chance that lesser mistakes will be made so it will be better to use the formula.