Problem 52 - Elementary Algebra

Question

Determine each of the values, \(|-2|^{3}\)

Step-by-Step Solution

Answer

Question: Determine the value of the given expression \(|-2|^{3}\). Answer: The value of the given expression \(|-2|^{3}\) is 8.

Step 1: Absolute Value

First, we need to find the absolute value of -2. Absolute value is a measure of the distance of a number from 0. So, regardless of the negative or positive sign of the number, its absolute value will be positive. In this case, the absolute value of -2 is 2: \(|-2| = 2\)

Step 2: Exponentiation

Now that we have found the base of our exponentiation operation (2), we can raise this base to the power of 3: \((|-2|)^{3} = 2^{3}\)

Step 3: Calculate the Result

We now need to multiply the base (2) by itself two more times (3 - 1 = 2 times). We have: \(2^{3} = 2 * 2 * 2 = 8\) Thus, the value of the given expression \(|-2|^{3}\) is 8.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers: a base and an exponent. It is a way to represent repeated multiplication. Here's how it works:

• The base is the number that is being multiplied.
• The exponent tells you how many times you multiply the base by itself.

For example, in the expression \(2^3\), 2 is the base and 3 is the exponent. This means you multiply 2 by itself three times: \(2 \times 2 \times 2\). This results in 8. Exponentiation allows for simplification of repetitive multiplication, making calculations quicker and easier to manage.

Base and Exponent

In any power expression, such as \(b^n\), there are two critical parts—the base \(b\) and the exponent \(n\).

The base is the number that you start with and repeatedly multiply. For example, in \(2^3\), 2 is the base. The exponent, found at the top right of the base, represents the number of times the base is used as a factor. For \(2^3\), 3 is the exponent, meaning you multiply the base, 2, three times: \(2 \times 2 \times 2\).

Understanding the roles of the base and exponent is crucial in simplifying and evaluating algebraic expressions involving powers. This knowledge helps when dealing with larger numbers and simplifies complex equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \(x\) or \(y\)), and arithmetic operations, such as addition, subtraction, multiplication, division, and exponentiation. They are used to convey relationships between quantities and can include constants—a fixed value—and variables, which can change.

Consider the expression \(3x^2 + 4y - 5\):

• \(3x^2\) implies 3 is a coefficient multiplying \(x\) squared.
• The term \(4y\) has a coefficient of 4 multiplying variable \(y\).
• -5 is a constant term added to the expression.

Such expressions can be simplified, evaluated, or solved for variable values under certain conditions. Mastery of algebraic expressions provides a foundation for higher-level math problems and real-world applications.