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Chapter 1: Problem 14

Write the expression in the form \(a+b i\), where a and \(b\) are real numbers. $$(4+7 i)(4-7 i)$$

Short Answer

Expert verified
The expression simplifies to \(65 + 0i\).

Step by step solution

01

Identify the Conjugate Pair

Notice that the expression \((4 + 7i)(4 - 7i)\) is a product of conjugates. The conjugate pairs are \(4 + 7i\) and \(4 - 7i\).
02

Use the Formula for Product of Conjugates

Apply the formula for the product of two conjugates: \((a + bi)(a - bi) = a^2 + b^2\). In this case, \(a = 4\) and \(b = 7\).
03

Calculate the Squares of the Real and Imaginary Parts

Compute \(a^2\) and \(b^2\) where \(a = 4\) and \(b = 7\). This yields \(a^2 = 4^2 = 16\) and \(b^2 = 7^2 = 49\).
04

Add the Results

Using the formula, add the results: \(a^2 + b^2 = 16 + 49 = 65\).
05

Write the Result as a Complex Number

Since the imaginary part cancels out, the result of \((4 + 7i)(4 - 7i)\) simplifies to \(65\), which is \(65 + 0i\) in the form \(a + bi\).

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Key Concepts

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

Conjugate Pairs
In the realm of complex numbers, **conjugate pairs** are quite interesting. A conjugate pair consists of two complex numbers with the same real part but opposite coefficients on the imaginary part. For example, given a complex number \( z = a + bi \), its conjugate would be \( \bar{z} = a - bi \).
  • The real parts in both numbers remain identical.
  • The imaginary parts have opposite signs.
This symmetry plays an important role in simplifying expressions involving complex numbers. In the exercise, the numbers \( 4 + 7i \) and \( 4 - 7i \) form a conjugate pair due to the similar real parts (4) and differing imaginary parts (+7 and -7). Conjugate pairs balance each other, often leading to simplifications when multiplied.
Product of Conjugates
A significant advantage of recognizing **product of conjugates** is simplifying complex expressions. The property of the product of conjugates formula is:
\[(a + bi)(a - bi) = a^2 + b^2\]When multiplying conjugate pairs, the middle term cancels out, resulting in a purely real number.
  • The middle term \((abi - abi)\) disappears.
  • What remains is the sum of the squares of the real and imaginary parts: \(a^2 + b^2\).
In our exercise involving \( 4 + 7i \) and \( 4 - 7i \), using this formula allows us to simplify the product quickly to \(16 + 49 = 65\). Multiplying conjugates is a powerful technique when the result needs to be expressed without an imaginary component.
Real and Imaginary Parts
Complex numbers consist of two parts: a real part and an imaginary part. For a complex number \( z = a + bi \), the real part is \(a\) and the imaginary part is \(b\), with \(i\) representing the square root of \(-1\).
  • Real parts can be found on the number line, straightforward and familiar like ordinary real numbers.
  • Imaginary parts involve \(i\), which introduces a new dimension in mathematics.
In calculations, as in this exercise, it's crucial to separate and treat these parts appropriately. The real part of our initial terms, 4, did not change; the manipulation of the imaginary part allowed for a simplified, real result. This shows that while imaginary components add complexity, they can also result in new truths when tackled correctly, such as the simplification of the expression to \(65 + 0i\).

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