Problem 32 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The simplified expression is -18.

Step 1: Simplify Exponentiation of Negative Base

Calculate \((-3)^3\) using the property of exponents. When a negative number is raised to an odd power, the result is negative. So, \((-3)^3 = (-3) imes (-3) imes (-3) = -27.\)

Step 2: Simplify Exponentiation of Positive Base

Calculate \(3^2\). Since the base is positive, this is simply \(3 imes 3 = 9.\)

Step 3: Add the Results

Combine the results of the exponentiations from Step 1 and Step 2.Add \(-27 + 9 = -18.\)

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They are usually represented with a minus sign in front. These can be a bit tricky because they behave differently than positive numbers in certain operations.
For example, multiplying two negative numbers results in a positive number. But, multiplying a negative number by a positive number gives a negative result. Here are a few points to remember:

• Negative × Negative = Positive
• Negative × Positive = Negative
• Negative raised to an odd power remains Negative
• Negative raised to an even power becomes Positive

Understanding these properties is essential when working with numerical expressions involving negative numbers.

Order of Operations

Order of operations is a set of rules that dictate how mathematical problems are solved. It determines the precise sequence to perform calculations and avoid confusion.
Simply put, it tells us what to do first in a numerical expression. The basic order is often remembered by the acronym PEMDAS:

• P – Parentheses first
• E – Exponents (i.e., powers and square roots, etc.)
• MD – Multiplication and Division (left-to-right)
• AS – Addition and Subtraction (left-to-right)

In the given exercise, this means evaluating exponents before anything else. Understanding this order is crucial for simplifying expressions correctly.

Numerical Expressions

A numerical expression is a combination of numbers and operations like addition, subtraction, multiplication, and division. It doesn't have an equal sign like an equation and can be simplified to a single number.
To handle these expressions effortlessly, follow these key tips:

• Identify all the operations involved (e.g., addition or multiplication)
• Apply the order of operations to solve
• Combine like terms where necessary

For the given expression \( (-3)^{3}+3^{2} \), applying exponents first was necessary to break down each part before adding the results. This approach ensures accuracy and simplicity.

Simplification

Simplification is the process of reducing a numerical expression to its simplest form. It typically involves performing all indicated operations within the expression to reach a single number.
A simplified expression is easier to read and understand, and it provides a clear answer.

• Start by resolving the exponents, as they represent repeated multiplication.
• Next, follow other operations adhering to the prescribed order.
• Combine numbers step-by-step until you reach the simplest form.

In our exercise, simplifying involved calculating each exponent first and then adding the results, resulting in an easy-to-understand final answer of \( -18 \). Understanding simplification helps in solving more complicated mathematical problems with ease.