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Chapter 3: Problem 67

Multiply as indicated. Write each product in standand form. $$i(3-4 i)(3+4 i)$$

Short Answer

Expert verified
The product is in standard form: 0 + 25i.

Step by step solution

01

Express complex conjugates multiplication

First, observe that the expression \((3 - 4i)(3 + 4i)\) is a multiplication of conjugates. Recall that \((a - bi)(a + bi) = a^2 + b^2\). So, set \(a = 3\) and \(b = 4\). We can apply the formula here.
02

Calculate the multiplication of conjugates

Using the formula, calculate \((3^2 + 4^2)\). This becomes \(9 + 16 = 25\). Thus, \((3 - 4i)(3 + 4i) = 25\).
03

Multiply the result with i

Now, multiply the result \(25\) by \(i\). This yields \(25i\).
04

Write the result in standard form

The standard form for complex numbers is \(a + bi\). Hence, \(25i\) can be written as \(0 + 25i\).

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Key Concepts

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

Complex Conjugates
The concept of complex conjugates is helpful in simplifying expressions, such as the one in the original exercise. A complex conjugate is a pair of complex numbers. If your complex number is in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, its complex conjugate will be \(a - bi\).

When you multiply a complex number by its conjugate, the result is always a real number. This happens because the imaginary parts cancel each other out. In mathematical terms, this is expressed as \((a+bi)(a-bi) = a^2 + b^2\).

  • To find the conjugate of \(3 - 4i\), you switch the sign of the imaginary part, giving \(3 + 4i\).
  • Multiplying \(3 - 4i\) and \(3 + 4i\), the result is a real number: \(25\).
Standard Form
Complex numbers are often expressed in what's known as standard form. This form is \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

Standard form is useful for easily identifying the real and imaginary components of a complex number. It is particularly important when you need to perform operations like addition, subtraction, or in this case, multiplication.

  • In the original exercise, the expression \(i(3-4i)(3+4i)\) initially results in \(25i\).
  • Since \(25i\) lacks a real component, it can be written in standard form as \(0 + 25i\).
Imaginary Unit
The imaginary unit, represented as \(i\), is one of the fundamental elements of complex numbers. It is defined by the property that \(i^2 = -1\). This unique attribute is what allows complex numbers to express quantities that real numbers alone cannot.

In the original exercise, \(i\) plays a pivotal role. It acts as a coefficient to our result after multiplying complex conjugates. When \(25\) is multiplied by \(i\), we are using the fundamental property of \(i\) to convert our real result into an imaginary number.

  • The operation transitions \(25\) into \(25i\), illustrating \(i\)'s ability to represent a purely imaginary number.
  • This showcases the significance of \(i\) in expanding the scope of mathematical operations beyond just real numbers.

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Most popular questions from this chapter

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(-x^{2}-x \leq 0\) (b) \(-x^{2}-x>0\)

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$P=\frac{E^{2} R}{(r+R)^{2}} \text { for } R$$

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(x-3 x^{2}>-1\) (b) \(x-3 x^{2} \leq-1\)

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$s=s_{0}+g t^{2}+k \quad \text { for } t$$

Explain why the method of dividing complex numbers (that is, multiplying both the numerator and the denominator by the conjugate of the denominator) works. What property justifies this process?
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