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Chapter 7: Problem 30

Find each product. Express each answer in the form \(a+b i\) $$(2+6 i)(2-6 i)$$

Short Answer

Expert verified
40

Step by step solution

01

鈥� Apply the FOIL Method

Use the FOIL method to multiply the binomials: First: Multiply the first terms: (2)(2) = 4 Outer: Multiply the outer terms: (2)(-6i) = -12i Inner: Multiply the inner terms: (6i)(2) = 12i Last: Multiply the last terms: (6i)(-6i) = -36i^2
02

鈥� Combine Like Terms

Combine the real parts and the imaginary parts separately: 4 + (-12i + 12i) + (-36i^2)
03

鈥� Simplify the Imaginary Unit

Recall that i^2 = -1, substitute -36i^2 with -36(-1): 4 + 0 + 36 = 4 + 36 = 40
04

鈥� Write the Final Answer

The final answer in the form of a + bi is: 40 + 0i which simplifies to 40

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Key Concepts

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

FOIL method
The FOIL method is a useful technique for multiplying two binomials. It stands for First, Outer, Inner, and Last, representing the terms you need to multiply:
  • First: Multiply the first terms in each binomial.
  • Outer: Multiply the outer terms in the binomials.
  • Inner: Multiply the inner terms.
  • Last: Finally, multiply the last terms in each binomial.
For example, in the product (2+6i)(2-6i) , you would perform the following multiplications:
  • First: (2)(2) = 4
  • Outer: (2)(-6i) = -12i
  • Inner: (6i)(2) = 12i
  • Last: (6i)(-6i) = -36i^2
Using the FOIL method helps you systematically approach the problem and ensures you don't miss any terms.
Combining like terms
Once you've used the FOIL method to multiply the binomials, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our example, after applying the FOIL method, you get: 4 + (-12i) + 12i - 36i^2 . We need to combine the like terms:
  • Combine the real constants: 4
  • Combine the imaginary terms: -12i + 12i , which cancel each other out to become 0.
We will then have: 4 + 0 - 36i^2 . This process of combining like terms simplifies our expression, which makes it easier to handle the next steps.
Imaginary unit properties
When dealing with complex numbers, it's important to understand the properties of the imaginary unit, denoted as i . The imaginary unit has a unique property: i^2 = -1 . This property is crucial when simplifying expressions involving complex numbers. In the example multiplication problem, after applying the FOIL method and combining like terms, you have: 4 - 36i^2 . Using the imaginary unit property, we substitute i^2 with -1 :
  • -36i^2 = -36(-1) = 36
This simplifies the expression further to: 4 + 36 = 40 . Therefore, our final answer in the form a + bi is 40 + 0i , which simplifies to 40 .

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Most popular questions from this chapter

Determine whether each equation is an identity. Explain. a) \(\left(w^{2} x^{2}\right)^{1 / 2}=|w| \cdot|x|\) b) \(\left(w^{2} x^{2}\right)^{1 / 2}=|w x|\) c) \(\left(w^{2} x^{2}\right)^{1 / 2}=w|x|\)

Discussion Which of the following equations are identities? Explain your answers. a) \(\sqrt{9 x}=3 \sqrt{x}\) b) \(\sqrt{9+x}=3+\sqrt{x}\) c) \(\sqrt{x-4}=\sqrt{x}-2\) d) \(\sqrt{\frac{x}{4}}=\frac{\sqrt{x}}{2}\)

Simplify each expression. $$\sqrt[3]{-16 x^{4}}+5 x \sqrt[3]{54 x}$$

Simplify each expression. $$\sqrt[3]{3 x^{5} y^{7}}-\sqrt[3]{24 x^{5} y^{7}}$$

Simplify. $$3 \sqrt{m}(2 \sqrt{m}-6)$$
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