Problem 6 - College Algebra

Question

For the following exercises, evaluate the algebraic expressions. If \(y=x^{3}-2,\) evaluate \(y\) given \(x=i\)

Step-by-Step Solution

Answer

The value of \(y\) is \(-2 - i\).

Step 1: Substitute x in the expression

First, take the given expression for \(y\): \(y = x^{3} - 2\). Substitute \(x = i\) into this expression. This gives us \(y = (i)^{3} - 2\).

Step 2: Calculate i cubed

Recall the property of the imaginary unit \(i\), where \(i^2 = -1\). Therefore, \(i^3 = i^2 \cdot i = -1 \cdot i = -i\).

Step 3: Simplify the expression

Now, replace \(i^3\) with \(-i\) in the expression \(y = i^3 - 2\). This substitution results in: \(y = -i - 2\).

Step 4: Final result

We have simplified the expression fully with the given value of \(x = i\). Thus, the evaluated expression is \(y = -2 - i\).

Key Concepts

Imaginary Unit

The imaginary unit, designated by the symbol \(i\), is a fundamental concept in the study of complex numbers. It is defined as the square root of negative one, so \(i^2 = -1\). This is particularly useful in mathematics, as it allows us to extend the real number system to include solutions to equations that cannot be solved using only real numbers. For example, the equation \(x^2 + 1 = 0\) has no real roots, but by introducing the imaginary unit, we can solve it using complex numbers.Understanding the properties of the imaginary unit can help in simplifying complex expressions. Here are some key properties:

• \(i^3 = i^2 \cdot i = -1 \cdot i = -i\)
• \(i^4 = (i^2)^2 = (-1)^2 = 1\)
• These properties repeat in a cycle: \(i, -1, -i, 1, i, \ldots\)

Recognizing these powers of \(i\) helps streamline calculations when working with complex numbers.

Evaluating Expressions

Evaluating expressions involves substituting the given values into an algebraic expression and simplifying it accordingly. In the context of complex numbers, such as the exercise where \(y = x^3 - 2\) and \(x = i\), evaluating expressions requires careful use of the properties of the imaginary unit.Here's how you can efficiently evaluate expressions:

• Identify the variable to substitute and correctly substitute the given value.
• Apply algebraic simplification rules, here noting \(i^2 = -1\) to simplify powers of \(i\).
• Simplify step-by-step, ensuring each step follows from previous algebraic rules.

Accurate evaluation of expressions often relies on understanding key concepts like the imaginary unit and maintaining clarity throughout the calculation process.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They represent a fundamental language in algebra, allowing us to describe relationships and solve equations. In complex numbers, algebraic expressions can go beyond real numbers, integrating imaginary components.Consider the expression \(y = x^3 - 2\). Here, \(x\) is a variable, and substituting \(x = i\) introduces complexity beyond real numbers. To handle such expressions:

• Substitute variables carefully, given any conditions or specific values, such as \(x = i\).
• Utilize known mathematical properties, like \(i^2 = -1\), to simplify the expression correctly.
• Simplifying these expressions often involves strategic grouping and ordering of terms.

Working with algebraic expressions involving complex numbers helps build a deeper understanding of both linear and non-linear relationships.