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Chapter 1: Problem 50

In Exercises \(45-50,\) perform the indicated operation(s) and write the result in standard form. $$ 5 \sqrt{-8}+3 \sqrt{-18} $$

Short Answer

Expert verified
The result of performing the indicated operation is \(19\sqrt{2}i\)

Step by step solution

01

Identify and Convert to Imaginary form

Identify the negative square roots in the expression. They are \(\sqrt{-8}\) and \(\sqrt{-18}\). Convert these negative square roots into their imaginary form.\n We know that the square root of negative one \(\sqrt{-1}\) is \(i\).\nSo, express each negative square root with \(i\).\nFor example, \(\sqrt{-8} = \sqrt{8} \cdot \sqrt{-1} = 2\sqrt{2}i\). And \(\sqrt{-18} = \sqrt{18} \cdot \sqrt{-1} = 3\sqrt{2}i\).
02

Substitute imaginary values into the expression

Substitute the imaginary values obtained in Step 1 back into the original expression. So, \(5\sqrt{-8} + 3\sqrt{-18}\) becomes \(10\sqrt{2}i + 9\sqrt{2}i\)
03

Add the two expressions

Now, add the two expressions together: \(10\sqrt{2}i + 9\sqrt{2}i = 19\sqrt{2}i\)

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Key Concepts

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

Imaginary Numbers
Imaginary numbers are an intriguing part of mathematics. They arise when we encounter the square root of a negative number, something which cannot be found within the realm of real numbers. Instead, we use a special unit called "i" to denote the imaginary unit.
  • The core property of the imaginary unit is that \( i^2 = -1 \).
  • When we write \( \sqrt{-1} \), we essentially write it as \( i \).

This concept allows us to extend our number system so that square roots of negative numbers are possible. It might appear abstract, but imaginary numbers have practical uses in engineering and physics, especially in describing electrical circuits and wave functions. By embracing imaginary numbers, calculations involving negative under-radicals become not only possible but much simpler.
Square Roots
Square roots are the values that, when multiplied by themselves, yield the original number. While the square root of a positive number can be neatly expressed as another real number, the plot thickens with negative numbers.
  • For a positive number \( a \), \( \sqrt{a} \) is a real number.
  • For a negative number \( -a \), \( \sqrt{-a} \) is imaginary and is expressed using \( i \). For example, \( \sqrt{-8} = \sqrt{8} \times i \).

This switch from real to imaginary is crucial in mathematical operations. Recognizing when to use imaginary units allows us to process a broader range of numerical problems confidently, bringing elegance to complex equations.
Standard Form
The standard form for complex numbers combines real and imaginary components. A complex number is typically expressed as \( a + bi \), where \( a \) represents the real part and \( b \) represents the imaginary part.
  • In our example, the result is purely imaginary: \( 19 \sqrt{2} i \).
  • Even when the real part is zero, the expression fits the standard form; in this case, \( 0 + 19 \sqrt{2} i \).

This notation helps in simplifying and consistently expressing complex numbers, making arithmetic operations such as addition or multiplication more straightforward. It ensures that real and imaginary parts are easily recognizable and separable.
Algebraic Operations
Algebraic operations with complex numbers involve addition, subtraction, multiplication, and sometimes division, just like with real numbers. However, it's important to manage both real and imaginary components.
Let's look into the steps:
  • Addition and Subtraction: Combine only the corresponding parts (real with real, imaginary with imaginary).
  • In our exercise, we added \( 10\sqrt{2}i \) and \( 9\sqrt{2}i \) to get \( 19\sqrt{2}i \).
  • Multiplication: Use the distributive property and include \( i^2 = -1 \) to simplify correctly.

Understanding these operations helps in solving complex problems and ensures that complex numbers are handled in a clear and systematic manner. Algebraic manipulations adhere to traditional rules with additional considerations for imaginary units, facilitating efficient problem-solving.

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Most popular questions from this chapter

Things did not go quite as planned. You invested \(\$ 12,000\), part of it in stock that paid \(14 \%\) annual interest. However, the rest of the money suffered a \(6 \%\) loss. If the total annual income from both investments was \(\$ 680,\) how much was invested at each rate?

Will help you prepare for the material covered in the next section. $$\text { Multiply: }(7-3 x)(-2-5 x)$$

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