Problem 77 - Elementary and Intermediate Algebra

Question

Simplify. Write the result in the form \(a+b i .\) \(3-\sqrt{-64}\)

Step-by-Step Solution

Answer

The result is \( 3 - 8i \).

Step 1: Convert the Square Root of a Negative Number

The expression contains \( \sqrt{-64} \). We know that \( \sqrt{-64} = \sqrt{64} \times \sqrt{-1} \). Since \( \sqrt{64} = 8 \) and \( \sqrt{-1} = i \), it follows that \( \sqrt{-64} = 8i \).

Step 2: Substitute the Imaginary Number

Substitute \( 8i \) back into the original expression. The expression now becomes \( 3 - 8i \).

Step 3: Write in Standard Form

The expression is already in the form \( a + bi \) because \( 3 \) is the real part, representing \( a \), and \(-8i\) represents the imaginary part \( bi \). Therefore, the final expression is \( 3 - 8i \).

Key Concepts

Imaginary Unit

The imaginary unit, often denoted as \( i \), is a fundamental concept in complex numbers. It is defined as the square root of \(-1\). This means \( i^2 = -1 \), which forms the basis for dealing with complex numbers. In mathematical problems, the introduction of \( i \) allows us to work with numbers that otherwise wouldn’t be possible to handle using only real numbers.

When faced with square roots of negative numbers, we turn to the imaginary unit. For instance, while \( \sqrt{-64} \) doesn’t exist in the realm of real numbers, it can be expressed as \( 8i \) using the imaginary unit as shown in the given solution.

This way of handling square roots is crucial as it helps in expanding the number system to include all roots, providing more solutions to equations.

Standard Form of a Complex Number

The standard form of a complex number is expressed as \( a + bi \), where:

• \( a \) is the real part
• \( bi \) is the imaginary part, with \( b \) being a real number and \( i \) the imaginary unit

This form helps in identifying and differentiating between the real and imaginary components of a number. In the exercise, after simplifying, the complex number obtained is \( 3 - 8i \). Here, \( 3 \) represents the real number part (a), whereas \( -8i \) is the imaginary part (bi).

Writing complex numbers in this standard form is essential for performing operations like addition, subtraction, or comparison of complex numbers. It provides a clean and unambiguous way to present the answer clearly.

Square Roots of Negative Numbers

Traditionally, the square root function is defined only for non-negative numbers. This means that the square root of a negative number, like \(-64\), isn't handled within real numbers. To address this, we extend the concept using the imaginary unit.

For any negative number, \( -b \), the square root can be expressed using \( i \) as \( \sqrt{b}i \). In our problem, \( \sqrt{-64} \) becomes \( \sqrt{64} \times i \). We know that \( \sqrt{64} = 8 \), so \( \sqrt{-64} = 8i \).

This method helps us when simplifying or calculating expressions involving negative square roots and ensures that such expressions have meaningful solutions applicable in various mathematical contexts and problem-solving scenarios.