Problem 22 - Algebra 2 and Trigonometry

Question

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \sqrt{-49}-\sqrt{-16} $$

Step-by-Step Solution

Answer

The result is \(3i\).

Step 1: Express Each Square Root in Terms of 'i'

First, recognize that the square root of a negative number can be expressed in terms of the imaginary unit, \(i\), where \(i = \sqrt{-1}\). Thus, \(\sqrt{-49} = \sqrt{49} \cdot \sqrt{-1} = 7i\) and \(\sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i\).

Step 2: Write the Expression in Terms of 'i'

Now, substitute the expressions found in Step 1 for \(\sqrt{-49}\) and \(\sqrt{-16}\) into the original problem. This gives us: \(7i - 4i\).

Step 3: Simplify the Expression

Combine the imaginary numbers by subtracting \(4i\) from \(7i\). This results in \((7 - 4)i = 3i\).

Key Concepts

Complex Numbers

Complex numbers might sound intimidating, but they are simply numbers that combine a real part and an imaginary part. In mathematical terms, a complex number is expressed as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The imaginary part includes \( i \), which is defined as \( i = \sqrt{-1} \).
Here are a few key points to remember about complex numbers:

• They extend the real number system, which makes it possible to perform operations that don't have answers within the real numbers alone, such as taking the square root of a negative number.
• When plotted on a graph, the real part corresponds to the horizontal axis, and the imaginary part corresponds to the vertical axis.
• Complex numbers are essential in many fields, including engineering and physics, because they can represent waveforms, oscillations, and electrical circuits.

Square Roots of Negative Numbers

One fascinating aspect of complex numbers is the ability to address the square roots of negative numbers. Traditionally, taking the square root of a negative number isn't possible within the real number system. This is because no real number squared results in a negative number.
Here's why imaginary numbers and \( i \) are essential:

• Any negative number under a square root can be expressed as a multiplication of its positive counterpart and \( \sqrt{-1} \). For instance, \( \sqrt{-49} = \sqrt{49} \cdot \sqrt{-1} = 7i \).
• Recognizing that \( \sqrt{-1} = i \) allows mathematicians and students to simplify and work with these expressions effortlessly.
• This adaptation allows the extension of quadratic equations solutions into realms not previously possible with only real numbers.

Simplifying Expressions with 'i'

Once you’ve converted those pesky square roots of negative numbers into terms of \( i \), simplifying expressions becomes a breeze. Here's a step-by-step guide to handle expressions such as the given \( 7i - 4i \):

• **Identify the real and imaginary parts.** In the expression \( 7i - 4i \), both terms are made up of imaginary components \( 7i \text{ and } 4i \).
• **Perform arithmetic operations on these terms.** Treat \( i \) just like any other algebraic term. Subtract \( 4i \) from \( 7i \) by dealing only with the coefficients: \( 7 - 4 = 3 \).
• **Combine and simplify the expression.** The result is \( 3i \), a much cleaner and concise form.

Why is this important? Being able to handle and simplify expressions with \( i \) expands your mathematical toolbox, proving invaluable in complex calculations whether in academia or applied sciences. It ensures that you can solve a wider range of problems, with clarity and confidence.