Problem 14 - Algebra A Combined Function

Question

Evaluate each expression with the given replacement values. $$ x^{3} \text { when } x=-2 $$

Step-by-Step Solution

Answer

The value of the expression is -8.

Step 1: Understand the Expression

The expression we need to evaluate is \( x^3 \). This means we need to find the cube of the variable \( x \), which is \( x \times x \times x \).

Step 2: Substitute the Value

The problem gives the value \( x = -2 \). Substitute \(-2\) into the expression, replacing \( x \) with \(-2\). So it becomes \((-2)^3\).

Step 3: Calculate the Cubed Value

To find \((-2)^3\), multiply \(-2\) by itself three times: \((-2) \times (-2) \times (-2)\).

Step 4: Perform the Multiplication

First, calculate \((-2) \times (-2) = 4\) because multiplying two negative numbers yields a positive number. Then multiply the result by \(-2\) again: \(4 \times (-2) = -8\).

Step 5: Conclude the Evaluation

The result of \((-2)^3\) is \(-8\). This is the final value of the expression \( x^3 \) when \( x = -2 \).

Key Concepts

Substitution

Substitution is a fundamental concept used in mathematics when we want to evaluate expressions. It involves replacing variables in an equation with their given values. In our exercise, we were instructed to substitute the variable \( x \) with \(-2\). This means we essentially set \( x = -2 \) and proceed with evaluating the expression.

• Start by identifying the variable: Here, the variable is \( x \).
• Use the given replacement value: Replace \( x \) with \(-2\).
• Ensure to correctly change the symbol(s) wherever the variable appears in the expression.

By performing substitution correctly, we transformed the expression \( x^3 \) into \((-2)^3\). This crucial step sets the stage for further calculation.

Negative Numbers Multiplication

Multiplication involving negative numbers can sometimes be tricky, but with a simple rule, it becomes straightforward. When two numbers are multiplied, the result will have a sign based on the signs of the numbers being multiplied:

• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative
• Negative × Positive = Negative

In the expression \((-2)^3\), we first multiply \(-2\) by itself to get \((-2) \times (-2) = 4\) because the multiplication of two negatives results in a positive. Then, \(4\) is further multiplied by \(-2\), and since a positive times a negative gives a negative, we end up with \(-8\). Understanding these rules helps in evaluating expressions correctly.

Exponentiation

Exponentiation refers to the action of raising a number to the power of another number. It means repeated multiplication of a number by itself. In our example, we are dealing with \((-2)^3\), which implies multiplying \(-2\) by itself three times:

• First multiply: \((-2) \times (-2) = 4\)
• Second multiply: \(4 \times (-2) = -8\)

The "3" in \(x^3\) represents the exponent, indicating how many times the base \(-2\) is used as a factor in the multiplication. The outcome, \(-8\), signifies the cubed power of \(-2\), which is the essence of exponentiation.

Evaluating Expressions

Evaluating expressions involves performing all the necessary mathematical operations to find the value of an expression. This process uses substitution, multiplication, and exponentiation principles. To evaluate \(x^3\) for \(x = -2\), we first substitute \(x\) with \(-2\), leading to the expression \((-2)^3\). Following substitution, utilize the rules of exponentiation, repeating the multiplication of \(-2\) three times.

• Substitute: \(x = -2\)
• Cubed value: Calculate \((-2) \times (-2) \times (-2)\)
• Compute: \((-2) \times (-2) = 4\), then \(4 \times (-2) = -8\)

This systematic evaluation process ensures accuracy, leading to the final result of \(-8\). Understand that each step plays a critical role in simplifying and solving mathematical expressions effectively.