Problem 31 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression simplifies to -17.

Step 1: Simplify the first term

Let's start by evaluating the expression \((-2)^{3}\). This means we need to multiply -2 by itself three times: \(-2 \times -2 \times -2\). First, calculate \(-2 \times -2 = 4\). Next, multiply the result by -2 again: \(4 \times -2 = -8\). So, \((-2)^{3} = -8\).

Step 2: Simplify the second term

Now, let's evaluate \(3^{2}\), which means multiplying 3 by itself: \(3 \times 3 = 9\). So, \(3^{2} = 9\).

Step 3: Subtract the two results

Now that we have simplified both terms, \((-2)^{3} = -8\) and \(3^{2} = 9\), we need to subtract the second result from the first result: \(-8 - 9\).Perform the subtraction: \(-8 - 9 = -17\).

Key Concepts

Numerical Expressions

Numerical expressions are composed of numbers and operations such as addition, subtraction, multiplication, and division. They are essentially math sentences that do not include variables. These expressions help us compute specific results using given numbers and operators.
To solve a numerical expression, it's important to:

• Understand each component, such as numbers and operation signs.
• Know what operations need to be performed and in which order.

In the expression \((-2)^{3} - 3^{2}\), we have two separate terms, \((-2)^{3}\) and \(3^{2}\), each of which is a numerical expression. Our job is to find the result of each before combining them.
Working with numerical expressions is a fundamental skill, providing a foundation for more complex math concepts, like algebra.

Simplification

Simplification is the process of making an expression easier to work with. This usually involves performing basic arithmetic operations to reach a single value or a simpler form.
When simplifying the expression \((-2)^{3} - 3^{2}\), we must break it down into smaller steps:

• First, evaluate each power: \((-2)^{3}\) and \(3^{2}\). This involves exponentiation, which means multiplying a number by itself a specific number of times.
• For \((-2)^{3}\), you multiply \(-2\) by itself three times: \(-2 imes -2 imes -2\), giving -8. Notice how the negative sign impacts the final result.
• For \(3^{2}\), multiply \(3\) by itself: \(3 \times 3\), which simplifies to 9.

After calculating these separately, \((-2)^{3} = -8\) and \(3^{2} = 9\), subtract the results as our final step in simplification: \(-8 - 9 = -17\). The art of simplification helps reduce complex expressions to straightforward answers, making them easier to understand and work with.

Order of Operations

The order of operations is a rule that describes the sequence in which different operations should be performed to accurately solve expressions. This sequence is crucial in mathematics to ensure consistency and accurate results.
In our problem \((-2)^{3} - 3^{2}\), the order of operations is essentially guiding how to handle the powers and subtraction:

• The first priority is exponentiation (evaluating powers), as seen with \((-2)^{3}\) and \(3^{2}\). Powers come before multiplication, division, addition, and subtraction.
• Once the exponentiation is completed, we handle subtraction, which is the final operation needed to simplify the expression.

The common mnemonic "PEMDAS" can help remember the order:

• P: Parentheses first
• E: Exponents (i.e., powers and roots, such as \(x^{2}\)
• MD: Multiplication and Division (left to right)
• AS: Addition and Subtraction (left to right)

Following the order of operations ensures that expressions are simplified correctly, leading to consistent and reliable results in mathematical computations.