Problem 6 - College Algebra with Corequisite Support

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 \(-i - 2\).

Step 1: Understanding the Expression

We start with the algebraic expression for \(y\), which is \(y = x^3 - 2\). Our task is to evaluate \(y\) when \(x = i\).

Step 2: Substitute the Value of x

Substitute \(x = i\) into the expression: \(y = i^3 - 2\).

Step 3: Calculate the Cube of i

Recall that \(i\) is the imaginary unit with the property that \(i^2 = -1\). Therefore, \(i^3 = i^2 \times i = (-1) \times i = -i\).

Step 4: Simplify the Expression

Now substitute \(-i\) back into the expression: \(y = -i - 2\).

Step 5: Conclusion

The value of \(y\) is \(-i - 2\), where \(-i\) is the imaginary part and \(-2\) is the real part.

Key Concepts

Algebraic Expressions

Algebraic expressions are fundamental elements in mathematics that involve variables, constants, and arithmetic operations such as addition, subtraction, multiplication, and division. In the exercise, we are working with the expression \(y = x^3 - 2\), where \(x\) is a variable that can be replaced by a specific value, in this case, the imaginary unit \(i\).

Evaluating algebraic expressions involves substituting variables with given values and performing the necessary calculations. This is crucial to solving problems and finding specific numerical results. For example, by substituting \(x = i\) into our expression, we calculate \(y = i^3 - 2\). To proceed, we need to understand how to work with the imaginary unit, \(i\).

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental element in the study of complex numbers. It is defined by the property that \(i^2 = -1\). This unique property allows us to extend real numbers to the complex number system, which includes all numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers.

When dealing with powers of \(i\), it helps to remember specific results:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = i^2 \times i = -1 \times i = -i\)
• \(i^4 = (i^2)^2 = (-1)^2 = 1\)

These values repeat in a cycle, which simplifies computations involving powers of \(i\). For this exercise, we used \(i^3 = -i\) to substitute back into the expression, getting \(y = -i - 2\).

Evaluation of Expressions

Evaluation of expressions involves a systematic approach to finding the value of an expression for given variable values. After substituting \(x = i\), we evaluated our expression \(y = i^3 - 2\).

Here's how the process unfolded:

• First, calculate the cube of \(i\), yielding \(-i\).
• Substitute \(-i\) back into the equation: \(y = -i - 2\).
• Identify the components: \(-i\) as the imaginary part and \(-2\) as the real part.

The final result is a complex number \(-i - 2\), demonstrating the evaluation process is not only for real numbers but also for complex expressions. Understanding how to handle these steps builds your proficiency in working with complex numbers and their expressions, enhancing your problem-solving skills.