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Chapter 8: Problem 454

Multiply using the Product of Complex Conjugates Pattern. \((6-5 i)(6+5 i)\)

Short Answer

Expert verified
The product is 61.

Step by step solution

01

Identify the Complex Conjugates

The complex numbers given are \(6 - 5i\) and \(6 + 5i\). When multiplying complex conjugates, we use the pattern \(a^2 - b^2\). Here, \(a = 6\) and \(b = 5i\).
02

Apply the Product of Complex Conjugates

The formula for the product of complex conjugates is \(a^2 - (bi)^2\). Substituting \(a = 6\) and \(b = 5i\), we get: \( (6)^2 - (5i)^2 \).
03

Simplify the Expression

Simplify each part of the expression: \( (6)^2 = 36 \) and \( (5i)^2 = 25i^2 \). Since \(i^2 = -1\), \( 25i^2 = 25(-1) = -25 \). So the expression becomes: \[ 36 - (-25) = 36 + 25 \].
04

Calculate the Final Result

Add \(36\) and \(+25\) to get the final result: \(36 + 25 = 61 \).

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Key Concepts

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

Complex Conjugates
Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. For example, if you have the complex number \(a + bi\), its conjugate is \(a - bi\). Conjugates are useful because their product always results in a real number. This is due to the fact that when you multiply \((a + bi)(a - bi)\), the imaginary parts cancel each other out. This property helps simplify complex number multiplication significantly.
Product of Complex Numbers
To multiply complex numbers, you generally use the distributive property (also known as FOIL method for binomials). However, multiplying complex conjugates is simpler. When you multiply \(a + bi\) by \(a - bi\), you use the pattern \(a^2 - b^2\). When these numbers are complex, this pattern still holds true but the \b^2\ term includes \(i^2\), the square of the imaginary unit. For example, multiplying \((6 - 5i)(6 + 5i)\) translates into \[ 6^2 - (5i)^2 \]. This gives \[ 36 - 25i^2 \] which simplifies to \[ 36 + 25 \] because \(i^2 = -1\). The final result is a real number: 61.
Imaginary Unit
The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined by the property that \(i^2 = -1\). This allows for the extension of the real number system to include solutions to equations like \(x^2 + 1 = 0\). When working with complex numbers, it's important to remember this definition because it helps to simplify expressions involving imaginaries. For instance, when you see something like \((5i)^2\), you know that \((5i)^2 = 25i^2 \) and since \((i^2 = -1)\), it simplifies further to \(-25\).

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Most popular questions from this chapter

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