Problem 48 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ (-2)^{3}+2(-2)^{2}-3(-2)-1 $$

Step-by-Step Solution

Answer

The simplified expression is 5.

Step 1: Simplify the Terms Inside the Parentheses

First, evaluate any exponentiation operations within the parentheses. Calculate \((-2)^3\): \((-2) imes (-2) imes (-2) = -8\). Calculate \((-2)^2\): \((-2) imes (-2) = 4\).

Step 2: Substitute and Apply Multiplication

Substitute the values found in Step 1 back into the original expression:\(-8 + 2 imes 4 - 3 imes (-2) - 1\). Now, compute the multiplication:- \(2 imes 4 = 8\)- \(-3 imes (-2) = -6\).

Step 3: Simplify the Expression

Substitute the multiplication results back and simplify:\(-8 + 8 - (-6) - 1\). Next, simplify further by performing each of the arithmetic operations step-by-step:First, \(-8 + 8 = 0\).

Step 4: Simplify Further by Handling Signs

Continue simplifying: Remember that subtracting a negative is the same as adding: \(0 - (-6)\) becomes \(0 + 6 = 6\).Finally, compute \(6 - 1 = 5\).

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is a foundational skill in mathematics. An algebraic expression is a combination of numbers, variables, and arithmetic operations.
For example, in the expression $$ (-2)^{3} + 2(-2)^{2} - 3(-2) - 1 $$, each term contains numbers, and some involve the mathematical operation of exponentiation.
The terms in this expression are independent units that can be individually evaluated and then combined according to the rules of algebra. In algebra:

• Variables represent unknown values. In this exercise, it's mainly about numbers, but many expressions incorporate variables.
• Constants are fixed numbers, like \(-2\) and \(2\) in our example.
• Operators such as addition (+), subtraction (-), and multiplication (*) are used to connect the numbers and variables.

Understanding how to read and transform these expressions is essential for solving algebraic problems, as it allows you to systematically approach finding a solution.

Simplification

Simplification in algebra involves reducing expressions to their simplest form, which often makes them easier to work with.
Simplification helps us understand and solve the problem more efficiently and can involve several operations like combining like terms, applying arithmetic operations, or performing operations such as exponentiation. Let's break down the simplification process:

• First, perform any operations within parentheses, which in our exercise includes calculating powers like \((-2)^3\) and \((-2)^2\).
• Replace the calculated values back into the expression. For instance, the cube of \(-2\) is \(-8\) and the square is \(4\).
• Solve any multiplication operations next, such as \(2 \times 4\) which equals \(8\).
• Finally, address any additions and subtractions sequentially. This includes handling signs, such as recognizing that subtracting a negative number is equivalent to adding its absolute value.

Simplifying step-by-step ensures accuracy and leads to the correct final result.

Arithmetic Operations

Arithmetic operations are basic calculations that include addition, subtraction, multiplication, and division.
In our context, they are used within algebraic expressions to evaluate and simplify terms. Here's a closer look at how they come into play:

• **Exponentiation**: In $$ (-2)^{3} $$, exponentiation is used. You multiply \(-2\) by itself three times, resulting in \(-8\).
• **Multiplication and Division**: Multiplication is applied after evaluating exponents, as seen in the term \(2(-2)^2\). Calculated as \(8\), the product impacts the expression's outcome greatly.
• **Addition and Subtraction**: These operations are often performed last, such as when simplifying \(-8 + 8\) leading to \(0\), then resolving \(0 - (-6)\) as \(0 + 6 = 6\).

Each of these operations has specific rules and precedences that govern their application, crucial for correctly solving the expression and ensuring a precise result.