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Chapter 12: Problem 48

Simplify each expression. $$\sqrt{-4} \sqrt{-4}$$

Short Answer

Expert verified
The simplified expression is -4.

Step by step solution

01

Rewrite with Imaginary Numbers

When dealing with square roots of negative numbers, we redefine them using the imaginary unit, denoted as "i," where \( i = \sqrt{-1} \). So, rewrite \( \sqrt{-4} \) as \( \sqrt{-1 \times 4} = \sqrt{-1} \times \sqrt{4} = 2i \).
02

Multiply the Imaginary Numbers

Now that we have expressed \( \sqrt{-4} \) as \( 2i \), we can multiply these expressions together:\( (2i) \times (2i) = 4i^2 \).
03

Simplify using the Property of i

Recall that \( i^2 = -1 \). Use this property to simplify the expression:\( 4i^2 = 4(-1) = -4 \).

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Key Concepts

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

Imaginary Unit
When we encounter square roots of negative numbers, the concept of imaginary numbers helps us define these expressions. The imaginary unit is symbolized by the letter "i." This unit is special because it represents the square root of negative one, expressed as \( i = \sqrt{-1} \).

The foundation of complex numbers is based on this unit, which allows us to extend the real number system to include these imaginary components. This concept is crucial when dealing with roots of negative numbers.
  • Example: \( \sqrt{-4} = \sqrt{4 \times -1} = 2i \).
  • Thus, whenever you see a square root of a negative number, think of the imaginary unit \( i \).
Properties of i
The imaginary unit \( i \) holds unique properties that simplify complex expressions. One of the primary properties is that \( i^2 = -1 \). This relation may seem strange, but it is essential when working with complex numbers. Here are some other important properties:
  • \( i^3 = i \times i^2 = i \times (-1) = -i \)
  • \( i^4 = i \times i^3 = i \times (-i) = 1 \)
  • These properties repeat every four powers, making computations predictable.
Understanding these properties can help you quickly simplify expressions that involve powers of \( i \). By using these characteristics, you transform what appears to be complex into something more manageable.
Simplification of Expressions
Using the imaginary unit \( i \), you can simplify mathematical expressions that involve square roots of negative numbers. The simplification process involves both rewriting the expression in terms of \( i \) and applying the properties of \( i \).

For example, when simplifying \( \sqrt{-4} \cdot \sqrt{-4} \), we first express each square root as \( 2i \). Multiplying these, \( (2i) \cdot (2i) = 4i^2 \).
  • Now, apply the property \( i^2 = -1 \) to get \( 4(-1) \).
  • The result is \( -4 \), a real number that simplifies the expression
    from a product of complex numbers back to a straightforward result.
Grasping the steps and understanding when and how to apply these adjustments simplifies calculations and deepens your mastery of complex numbers.

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Most popular questions from this chapter

Determine the partial fraction decomposition for each of the given expressions. $$\frac{p x+q}{(x-a)(x+a)} \quad(a \neq 0)$$

Determine whether the given value is a zero of the function. \(f(x)=x^{3}-3 x^{2}+3 x-3\) (a) \(x=\sqrt[3]{2}-1\) (b) \(x=\sqrt[3]{2}+1\)

Find a quadratic equation with the given roots. Write your answers in the form \(A x^{2}+B x+C=0\) Suggestion: Make use of Table 2. $$r_{1}=1+i \sqrt{3}, r_{2}=1-i \sqrt{3}$$

Determine the partial fraction decomposition for each of the given expressions. $$\frac{1}{(x-a)(x+a)} \quad(a \neq 0)$$

First graph the two functions. Then use the method of successive approximations to locate, between successive thousandths, the \(x\)-coordinate of the point where the graphs intersect.Use a graphing utility to draw the graphs as well as to check your final answer. $$y=x^{3} ; y=4-x^{2}$$
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