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Chapter 1: Problem 34

Evaluate the product, and write the result in the form \(a+b i\) $$(3-7 i)(3+7 i)$$

Short Answer

Expert verified
The result is \(58\).

Step by step solution

01

Identify the Pattern

The expression \((3 - 7i)(3 + 7i)\) is a product of two conjugates. In general, the product of conjugates \((a + bi)(a - bi)\) is equal to \(a^2 + b^2\).
02

Apply the Formula

Using the formula for the product of complex conjugates, compute: \[ (3)^2 + (7)^2 = 9 + 49 \]
03

Simplify the Expression

Add the results from Step 2: \[ 9 + 49 = 58 \]
04

Write the Result in the Required Form

Since the imaginary part is zero, the final result in the form \(a + bi\) is \(58 + 0i\).

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Key Concepts

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

Conjugate Pairs
In the world of complex numbers, conjugate pairs hold significant importance. A conjugate pair is a set of two complex numbers, where one is the mirror image of the other across the real axis.
In mathematical terms, for a complex number in the form of \(a + bi\), its conjugate is\(a - bi\).
  • The purpose of using conjugate pairs is to simplify expressions, especially when it comes to calculation and solving equations.
  • Multiplying a complex number by its conjugate results in a real number. This magic happens because the imaginary parts cancel each other out.
For example, the product of \((3-7i)\) and \((3+7i)\) can easily be evaluated to a real number because they are conjugate pairs.
By using the formula \((a + bi)(a - bi) = a^2 + b^2\), complex conjugates turn potentially tricky calculations into straightforward operations.
Complex Multiplication
Complex multiplication involves both real and imaginary numbers. When multiplying two complex numbers, \((a + bi)(c + di)\), we use the distributive property, similar to multiplying binomials.
Here's the step-by-step approach:
  • Multiply the real parts: \(a \times c\)
  • Multiply the imaginary parts: \(b \times d\ i^2\) (Recall that \(i^2 = -1\))
  • Multiply the real with the imaginary: \(a \times d\ i\)
  • Multiply the imaginary with the real: \(b \times c\ i\)
Add these products together to obtain a final result.
In our exercise, \((3 - 7i)(3 + 7i)\), we see how conjugate pairs simplify this multiplication.
The imaginary components vanish, resulting in the real number \(58\) after using our conjugate formula, making complex multiplication much clearer in specific scenarios.
Imaginary Numbers
Imaginary numbers form the foundation of complex numbers. They are essential for solving equations that lack real number solutions, such as square roots of negative numbers.
Imaginary numbers are denoted by \(i\), where \(i^2 = -1\). This unique property allows us to extend beyond the limitations of conventional numbers and opens up a broader field of mathematics known as complex analysis.
  • A single imaginary number is represented as \(bi\), where \(b\) is a real number coefficient.
  • Imaginary numbers, when combined with real numbers, form a complex number, \(a + bi\).
  • In mathematics and engineering, imaginary numbers are crucial for describing oscillations, waves, and alternating currents.
Though they might sound mystical, imaginary numbers are as valid as real numbers in various real-world applications, making them an indispensable component of mathematics.

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Most popular questions from this chapter

Test the equation for symmetry. $$x^{4} y^{4}+x^{2} y^{2}=1$$

Area of a Region Find the area of the region that lies outside the circle \(x^{2}+y^{2}=4\) but inside the circle $$ x^{2}+y^{2}-4 y-12=0 $$

DISCOVER - PROVE: Relationship Between Solutions and Coefficients The Quadratic Formula gives us the solutions of a quadratic equation from its coefficients. We can also obtain the coefficients from the solutions. (a) Find the solutions of the equation \(x^{2}-9 x+20=0\) and show that the product of the solutions is the constant term 20 and the sum of the solutions is \(9,\) the negative of the coefficient of \(x\) (b) Show that the same relationship between solutions and coefficients holds for the following equations:$$ \begin{array}{l}x^{2}-2 x-8=0 \\\x^{2}+4 x+2=0\end{array}$$ (c) Use the Quadratic Formula to prove that in general, if the equation \(x^{2}+b x+c=0\) has solutions \(r_{1}\) and \(r_{2}\) then \(c=r_{1} r_{2}\) and \(b=-\left(r_{1}+r_{2}\right)\)

Radicals Simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers. (a) \(\sqrt{4 s t^{3}} \sqrt[6]{s^{3} t^{2}}\) (b) \(\frac{\sqrt[4]{x^{7}}}{\sqrt[4]{x^{3}}}\)

Complete the following tables. What happens to the \(n\) th root of 2 as \(n\) gets large? What about the \(n\) th root of \(\frac{1}{2} ?\) $$\begin{array}{|r|r|}\hline n & 2^{1 / n} \\\\\hline 1 & \\\2 & \\\5 & \\\10 & \\\100 & \\\\\hline\end{array}$$ $$\begin{array}{|r|r|} \hline n & \left(\frac{1}{2}\right)^{1 / n} \\\\\hline 1 & \\\2 & \\\5 & \\\10 & \\\100 & \\\\\hline\end{array}$$ Construct a similar table for \(n^{1 / n}\). What happens to the \(n\) th root of \(n\) as \(n\) gets large?
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