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Chapter 3: Problem 50

Simplify. Write answers in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$(5+9 i)(5-9 i)$$

Short Answer

Expert verified
106 + 0i

Step by step solution

01

Recognize the form

Identify that the given expression (5+9i)(5-9i) is in the form (a+bi)(a-bi), which is the product of a complex number and its conjugate.
02

Apply the formula

Use the special product formula (a+bi)(a-bi)=a^2 - (bi)^2.
03

Substitute the values

Substitute a=5 and b=9 into the formula: (5+9i)(5-9i)=5^2 - (9i)^2.
04

Simplify the expression

Calculate each part of the equation: 5^2 = 25 and (9i)^2 = (9^2)(i^2) = 81(-1) = -81. Therefore, 25 - (-81) = 25 + 81.
05

Combine results

Add the results from the previous step: 25 + 81 = 106. Thus, the simplified form is 106 + 0i.

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Key Concepts

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

complex conjugate
In the world of complex numbers, every number has a special partner called its complex conjugate. If you have a complex number in the form of \(a+bi\), its conjugate would be \(a-bi\).
The complex conjugate is useful for simplifying complex number expressions, especially when dealing with multiplication. This is because when you multiply a complex number by its conjugate, you end up with a real number. For instance, in the exercise, the complex numbers were \(5+9i\) and its conjugate \(5-9i\).
When multiplied together, they eliminate the imaginary parts and give you a straightforward result.
Special Product Formula
One powerful tool in simplifying expressions with complex numbers is the special product formula. This formula is:
\((a+bi)(a-bi) = a^2 - b^2i^2\).
Notice how the imaginary parts \(i\) cancel out due to the properties of multiplication. This makes the formula a straightforward way to handle what might seem like complex multiplication.
In the exercise, the expression \((5+9i)(5-9i)\) matched this format perfectly. By applying the special product formula to it, we easily simplified it to a result of \(106+0i\), which is just the real number 106.
The special product formula helps streamline calculations and is a valuable shortcut when dealing with complex conjugates.

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