Problem 32 - Intermediate Algebra

Question

Write each number in the form a \(+b i.\) a. \(21+\sqrt{-16}\) b. \(-9+\sqrt{-12}\)

Step-by-Step Solution

Answer

a. \( 21 + 4i \), b. \( -9 + 2\sqrt{3}i \)

Step 1: Simplify the Square Root in (a)

First, identify the imaginary component of \( \sqrt{-16} \). Replace \( \sqrt{-16} \) by \( 4i \), because \( \sqrt{-16} = \sqrt{16} \times \sqrt{-1} = 4i \).

Step 2: Combine Real and Imaginary Parts in (a)

Now, write the expression \( 21 + \sqrt{-16} \) in the form \( a + bi \). Substitute \( 4i \) for \( \sqrt{-16} \), to get \( 21 + 4i \).

Step 3: Simplify the Square Root in (b)

For \( \sqrt{-12} \), find the imaginary component. Rewrite \( \sqrt{-12} \) as \( \sqrt{12} \times \sqrt{-1} = 2\sqrt{3}i \), because \( \sqrt{12} = 2\sqrt{3} \).

Step 4: Combine Real and Imaginary Parts in (b)

Rewrite the expression \( -9 + \sqrt{-12} \) in the form \( a + bi \). Substitute \( 2\sqrt{3}i \) for \( \sqrt{-12} \), so it becomes \( -9 + 2\sqrt{3}i \).

Key Concepts

Imaginary Numbers

In the realm of numbers, imaginary numbers can seem a bit mysterious at first. They come into play when we deal with the square roots of negative numbers. Typically, we know that taking the square root of a positive number is straightforward: for example, the square root of 9 is 3. However, what if we need to find the square root of -9? This is where imaginary numbers become very useful.

An imaginary number is expressed as multiples of 'i', where 'i' is defined as the square root of -1. Mathematically, it can be written as:

• \[ i = \sqrt{-1} \]
• \[ i^2 = -1 \]

So, when you take the square root of a negative number, it can be separated into a real number multiplied by 'i'. For example, \( \sqrt{-16} \) becomes \( 4i \), because:

• \[ \sqrt{-16} = \sqrt{16} \times \sqrt{-1} = 4 \times i = 4i \]

This separation allows us to handle these negative roots just like we handle positive roots, but with the added twist of using 'i'.

Algebra

Algebra is like a language of mathematics, allowing us to express and solve equations involving unknown values. In these original exercises, you were asked to write numbers in a specific format, known as complex numbers: `a + bi`.

Complex numbers consist of two parts:

• The real part, denoted as 'a', which is a regular integer or fraction.
• The imaginary part, denoted as 'bi', where 'b' is a real number multiplied by \( i \), the imaginary unit.

Writing numbers in this form allows for a neat way to handle both real and imaginary components together. For example, take the number \( 21 + 4i \). Here, 21 is the real part, and 4i is the imaginary part. Breaking it down like this keeps our calculations organized and makes interactions with other complex numbers much more manageable. Complex numbers are widely used in many fields of science and engineering, offering a robust framework for dealing with two-dimensional quantities.

Square Roots

Square roots play a fundamental role in mathematics, acting as the opposite of squaring a number. Finding the square root of a positive number returns a value which, when squared, gives the original number. For instance, the square root of 25 is 5 because \( 5^2 = 25 \).

When dealing with negative numbers, the process takes an interesting turn. Typically, the idea of taking the square root of a negative number was seen as impossible. However, by introducing imaginary numbers, we can handle these cases meaningfully. When you encounter a square root of a negative number, such as \( \sqrt{-12} \), it is expressed in terms of 'i', utilizing the relationship \( i = \sqrt{-1} \).
To simplify \( \sqrt{-12} \), we would first express it as \( \sqrt{12} \times \sqrt{-1} \), which simplifies further to \( 2\sqrt{3}i \). Understanding how to break down and reassemble these components lets us convert complex-looking expressions into manageable mathematical forms, which is essential for solving algebraic equations involving complex numbers.