Problem 59 - Algebra 2

Question

Simplify each expression. $$ (4+\sqrt{-9})+(6-\sqrt{-49}) $$

Step-by-Step Solution

Answer

\(10 - 4i\)

Step 1: Identify and Simplify Imaginary Numbers

Identify the square roots of negative numbers which represent imaginary numbers. Simplify \(\sqrt{-9} = 3i\) and \(\sqrt{-49} = 7i\), where \(i\) is the imaginary unit.

Step 2: Combine Real and Imaginary Parts

Combine the real parts and the imaginary parts separately. Real parts: \(4 + 6\). Imaginary parts: \(3i - 7i\).

Step 3: Perform the Addition

Perform the addition of real parts: \(4+6=10\). Perform the subtraction of imaginary parts: \(3i - 7i = -4i\).

Step 4: Write the Final Simplified Expression

Combine the simplified real and imaginary parts to get the final expression: \(10 - 4i\).

Key Concepts

Imaginary Unit

The concept of the imaginary unit is fundamental in mathematics, especially when dealing with complex numbers. It is defined as the square root of -1 and is denoted by the symbol i. This might seem like a strange idea at first, because all the numbers we're used to dealing with have real square roots. But in mathematics, we extend the number system to include this new type of number.

The power of the imaginary unit is that it allows us to write the square root of any negative number. For instance, the square root of -9 is written as 3i, because \( \(\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i\) \) where \(i = \sqrt{-1}\). Whenever you see the square root of a negative number, you can simplify it using the imaginary unit.

Complex Numbers

Complex numbers are made up of two parts: a real part and an imaginary part. They are written in the form a + bi, where a and b are real numbers, and i is the imaginary unit. The real part is a, and the imaginary part is bi. For instance, in the number 10 - 4i, 10 is the real part, and -4i is the imaginary part.

Understanding the structure of complex numbers is crucial when simplifying expressions that include both real and imaginary parts. The goal is to keep similar parts together - real with real, and imaginary with imaginary. By doing so, we can manage complex numbers in a way that is somewhat similar to how we handle regular algebraic operations with real numbers.

Simplify Expressions

Simplifying expressions, whether they include real numbers, variables, or complex numbers, involves reducing them to the simplest form possible. When simplifying expressions that contain imaginary numbers, as in our example, the approach is to first simplify each imaginary term. This could mean converting square roots of negative numbers into their imaginary form.

After simplifying the individual terms, the next step is to combine like terms. For a complex expression like \( (4+\sqrt{-9})+(6-\sqrt{-49}) \), we simplify it by grouping real numbers and imaginary numbers, then perform basic algebraic operations, such as addition and subtraction. This sort of organized approach makes managing complex expressions more manageable and leads to a neater, more understandable final result.

Algebraic Operations

Algebraic operations are the bread and butter of working with mathematical expressions, whether they represent real or complex numbers. These operations include addition, subtraction, multiplication, and division. When dealing with complex numbers, these operations are carried out separately on the real and imaginary parts of the numbers.

For example, when adding two complex numbers, you would add the real parts together and the imaginary parts together. Using our problem as an example, to simplify \( (4+\sqrt{-9})+(6-\sqrt{-49}) \) we added the real parts (4 and 6) to get 10, and then subtracted the imaginary parts (3i and 7i) to get -4i, resulting in a simplified expression of 10 - 4i. These algebraic operations follow the same rules as with real numbers, just carried out in two streams - one for the real component and one for the imaginary component.