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Chapter 10: Problem 69

Perform each indicated operation. Write the result in the form \(a+b i\) $$ (\sqrt{3}+2 i)(\sqrt{3}-2 i) $$

Short Answer

Expert verified
The result is \(7 + 0i\).

Step by step solution

01

Recognize the expression format

Observe that the expression \((\sqrt{3} + 2i)(\sqrt{3} - 2i)\) is in the form of \((a+bi)(a-bi)\), where \(a = \sqrt{3}\) and \(b = 2\). This is a difference of squares, which simplifies to \(a^2 - b^2\).
02

Apply the difference of squares

Use the identity \((a+bi)(a-bi) = a^2 - (bi)^2\). In this case, substitute \(a = \sqrt{3}\) and \(b = 2\) into the formula: \((\sqrt{3})^2 - (2i)^2\).
03

Calculate each square

Calculate \((\sqrt{3})^2 = 3\) and \((2i)^2 = 4i^2\). Recall that \(i^2 = -1\), so \(4i^2 = 4(-1) = -4\).
04

Simplify the expression

Simplify the expression using the results from Step 3: \((\sqrt{3})^2 - (2i)^2 = 3 - (-4)\). Therefore, the expression simplifies to \(3 + 4 = 7\).
05

Express in the form \(a + bi\)

Since the imaginary part is zero, express the result as \(7 + 0i\). This is in the form \(a + bi\), where \(a = 7\) and \(b = 0\).

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Key Concepts

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

Difference of Squares
The difference of squares is a powerful algebraic identity. It shows how to calculate the product of two conjugates. A conjugate pair can be written as
  • a + b and a - b
When multiplied, they result in another expression: \[(a+b)(a-b) = a^2 - b^2\] The trick is in recognizing this format quickly. In the problem, the
  • terms are conjugates: \((\sqrt{3} + 2i)(\sqrt{3} - 2i)\)
The difference of squares formula simplifies this to \((\sqrt{3})^2 - (2i)^2\).This strategy not only saves time but also makes the multiplication of complex numbers straightforward.
Imaginary Unit
In mathematics, the imaginary unit is represented by the symbol \(i\). It stands for the square root of
  • -1
and importantly, it holds some special properties: \[i^2 = -1\] As seen in the multiplication
  • of \(2i\)
by \(2i\), which results in
  • \((2i)^2 = 4i^2\).
Remember, whenever you square \(i\), it turns into -1. This feature helps in transforming imaginary numbers into real numbers during simplification.
Simplification
Simplification involves making an expression easier to understand.
  • The expression \((\sqrt{3})^2 - (2i)^2\) was simplified using basic arithmetic and properties of the imaginary unit.
You first evaluate each term:
  • \((\sqrt{3})^2 = 3\)
  • \((2i)^2 = 4i^2 = 4(-1) = -4\)
Next, you subtract these results:
  • 3 minus -4 equals 7.
Simplification frequently involves
  • combining like terms
and using identities to achieve a plainer form.
Algebraic Expressions
Algebraic expressions combine numbers, variables, and operations. They form the foundation of algebraic solutions. In our scenario with
  • complex numbers
like
  • \(\sqrt{3} + 2i\)
we use arithmetic rules to simplify. The operations include addition, subtraction, and division and must account for the imaginary unit. Key ideas include:
  • Recognizing familiar patterns (like difference of squares).
  • Handling the imaginary unit so it simplifies computations.
After simplification, the result retains the form \(a + bi\). Here, it becomes \(7 + 0i\). This underscores the scope of algebraic expression simplification.

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Most popular questions from this chapter

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(x^{3} y^{2}\right)^{1 / 4}}{\left(x^{-5} y^{-1}\right)^{-1 / 2}} $$

Use a calculator to write a four-decimal-place approximation of each number. $$ 20^{1 / 5} $$

Which of the following are not real numbers? Explain why \(\sqrt{-64}\) is not a real number.

Simplify each exponential expression. $$\frac{7 x^{-1} y}{14\left(x^{5} y^{2}\right)^{-2}}$$

Factor the common factor from the given expression. $$ x^{8 / 3} ; x^{8 / 3}+x^{10 / 3} $$
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