Multiply the following and simplify. ( 7 + 3 i ) ( 7 − 3 i ) \left(7+3i\right)\left(7-3i\right)

we're asked to multiply the following and simplify. We have seven plus 3i times seven minuteus 3i. So we have almost the same exact terms here being multiplied, but with different signs in the middle. So let's see what happens when we multiply here. Because we're multiplying two binomials, we want to go ahead and foil. So starting by multiplying those first terms, seven times seven will give me 49. Then our outside terms, seven times negative three I, that's going to be negative 21 I. Then our inside terms, three I times seven, that's going to be positive 21 I. And finally, those last terms, positive three I times negative three I, that's going to be negative nine I squared, since I have I times I happening there. Now the first thing I notice is I have negative 21 I and positive 21 I. Those are going to cancel each other out, just leaving me with 49 minus nine I squared. Now we can simplify further because we know that I squared is equal to negative one. So replacing that I squared with a negative one gives me here 49 minus nine times negative one. Negative nine times negative one is positive nine, so that makes this 49 plus nine, which gives us a final answer here of 58. So having multiplied these two terms that differ just by that sign in the middle, we ended up with just a real number integer value, 58. So let's keep that in mind as we continue on to our next lesson. I'll see you there.

Multiply the following and simplify. ( 7 + 3 i ) ( 7 − 3 i ) \left(7+3i\right)\left(7-3i\right)

49 − 9 i 2 49-9i^2 49 − 9 i 2

9. Roots, Radicals, and Complex Numbers

Multiplying and Dividing Complex Numbers

Intermediate Algebra

9. Roots, Radicals, and Complex Numbers

Multiplying and Dividing Complex Numbers

Struggling with Intermediate Algebra?

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Multiple Choice

Multiply the following and simplify.
$(7 + 3 i) (7 - 3 i)$

$58$

$40$

$49+9i$

$49-9i^2$

Verified step by step guidance

Recognize that the expression $ (7 + 3i)(7 - 3i) $ is a product of two complex conjugates. This is a special product that can be simplified using the difference of squares formula.

Apply the difference of squares formula: $ (a + b)(a - b) = a^2 - b^2 $. Here, $ a = 7 $ and $ b = 3i $, so rewrite the expression as $ 7^2 - (3i)^2 $.

Calculate each square separately: $ 7^2 = 49 $ and $ (3i)^2 = 9i^2 $.

Recall that $ i^2 = -1 $, so substitute this into the expression to get $ 49 - 9(-1) $.

Simplify the expression by multiplying $ -9 $ and $ -1 $ to get $ 49 + 9 $, then combine like terms to find the simplified result.

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Multiply the following and simplify.(7+3i)(7−3i)\left(7+3i\right)\left(7-3i\right)

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Multiply the following and simplify.
$(7 + 3 i) (7 - 3 i)$