Problem 22 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ 6 i-(4-i) $$

Step-by-Step Solution

Answer

-4 + 7i is the evaluated expression in the form a + bi.

Step 1: Distribute the Negative Sign

The expression given is \(6i - (4-i)\). First, distribute the negative sign inside the parentheses to remove them. This changes the expression to \(6i - 4 + i\).

Step 2: Combine Like Terms

Now, combine the like terms in the expression. Combine the imaginary parts \(6i\) and \(i\), resulting in \(7i\). Keep the real part \(-4\) as it is. The expression becomes \(-4 + 7i\).

Key Concepts

Imaginary Unit

The imaginary unit is a fundamental concept in the complex number system. It is denoted as \(i\), which represents the square root of \(-1\). This is a very unique number as no real number squared yields a negative product. Imaginary numbers are used mainly in engineering and physics.

• The core purpose of the imaginary unit is to extend the real number system to solve all polynomial equations, especially those that involve square roots of negative numbers.
• Imaginary numbers combined with real numbers form complex numbers, expressed in the standard form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part multiplied by \(i\).

Understanding \(i\) is essential as it helps in performing mathematical operations involving complex numbers, such as addition, subtraction, multiplication, and division. Always remember that in any expression, the square of \(i\) is \(-1\). This knowledge simplifies calculations and helps structure complex numbers into a form that can be easily manipulated.

Combine Like Terms

When working with algebraic expressions, particularly those that involve complex numbers, it's important to combine like terms. This means to add or subtract terms that have the same variable part.

• In the context of complex numbers, this involves separately handling the real and imaginary components.
• Only terms with \(i\) can be combined with other terms featuring \(i\), which is referred to as combining the imaginary parts of the expression.

For instance, in the given expression after distributing the negative sign — \(6i - 4 + i\) — you have to add \(6i\) and \(i\) because they share the imaginary unit \(i\). This results in \(7i\). The real number, \(-4\), does not have a like term to combine with, so it remains as is in the expression \(-4 + 7i\). This step is crucial for simplifying expressions into their standard complex form.

Distribute Negative Sign

Distributing a negative sign is a fundamental algebraic operation used to simplify expressions. It involves applying the negative sign outside the parentheses to each term inside the parentheses. This step is particularly important when dealing with subtraction.

• In the given expression, \(6i - (4 - i)\), distributing the negative sign allows us to simplify by changing the signs of the terms inside the parentheses. This gives \(6i - 4 + i\).
• After distribution, the operation within changes to addition and subtraction of individual terms without parentheses.

Always remember that incorrect distribution could lead to errors in solving your expression. It simplifies complex expressions and helps in the clear combination of like terms, ultimately yielding the expression in its simplest form, which here becomes \(-4 + 7i\). Making this step second nature will enhance your algebra skills and accuracy.