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Chapter 7: Problem 55

Multiply and simplify. Write each answer in the form \(a+b i\). $$(3+8 i)(3-8 i)$$

Short Answer

Expert verified
73

Step by step solution

01

Identify the form

The expression (3+8 i)(3-8 i) is in the form of (a+bi)(a-bi). Recognize that this is a product of two conjugates.
02

Apply the difference of squares formula

Use the formula (a+bi)(a-bi) = a^2 - (bi)^2. In this case, a=3 and b=8.
03

Calculate each term

First, calculate a^2: 3^2 = 9. Then, calculate (bi)^2: (8i)^2 = 64i^2. Note that i^2 = -1, so 64i^2 = 64(-1) = -64.
04

Subtract the terms

Now subtract the terms found in the previous step: 9 - (-64) = 9 + 64.
05

Simplify the expression

Combine the terms to get the final simplified form: 9 + 64 = 73.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

difference of squares
In mathematics, the difference of squares is a very useful formula. It helps to simplify products of certain kinds of expressions. The formula is \[ (a + b)(a - b) = a^2 - b^2 \]. This formula tells us that when we multiply two binomials where one is the sum and the other is the difference of the same two terms, the result is the difference of their squares.
For example, in our exercise, we have the expression \[ (3 + 8i)(3 - 8i) \]. Here, we can see that the first binomial is the sum of 3 and 8i, and the second binomial is the difference of 3 and 8i. By using the difference of squares formula, we can quickly simplify the multiplication.
complex conjugates
Complex conjugates are pairs of complex numbers that only differ by the sign of their imaginary part. So, for a complex number \(a + bi\), its conjugate is \(a - bi\). These conjugates have a special property: when multiplied together, they give a real number. This property makes complex conjugates very useful for simplifying expressions involving complex numbers.
In our exercise, \(3 + 8i\) and \(3 - 8i\) are complex conjugates. Multiplying them together results in a difference of squares, allowing us to eliminate the imaginary parts and simplify to a real number.
simplifying expressions
Simplifying expressions is a critical skill in algebra and calculus. For complex numbers, it involves reducing the expression to its simplest form, often by eliminating imaginary parts where possible.
In the given exercise, we use the difference of squares and the properties of complex conjugates to simplify the expression \[(3+8i)(3 - 8i)\]. By applying the difference of squares formula, we get \[3^2 - (8i)^2\]. Simplifying further, we calculate \[ 3^2 = 9\] and \[(8i)^2 = 64i^2\]. Recall that \(i^2 = -1\), so \[64i^2 = 64(-1) = -64\]. Finally, we subtract the terms: \[9 - (-64) = 9 + 64 = 73\]. Thus, the simplified answer is 73, a real number.

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