Problem 27 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is -19.

Step 1: Calculate Powers

First, calculate the powers in the expression. Compute \(2^3\) and \(3^3\). \(2^3 = 2 \times 2 \times 2 = 8\)\(3^3 = 3 \times 3 \times 3 = 27\)

Step 2: Subtract the Results

Now subtract the result of \(3^3\) from \(2^3\).\(8 - 27 = -19\)

Key Concepts

Exponentiation

Exponentiation is an arithmetic operation that involves raising a base number to a certain power, referred to as an exponent. This operation has its own rules and is often used to simplify expressions. It’s important to understand how it works:

• The base is the number that gets multiplied.
• The exponent is the number that indicates how many times the base is multiplied by itself.

When you see something like \(2^3\), it means you multiply the base \(2\) by itself \(3\) times, which results in \(2 \times 2 \times 2 = 8\). Similarly, \(3^3\) is \(3 \times 3 \times 3 = 27\).
One crucial aspect of exponentiation in mathematical expressions is the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Exponentiation is executed before multiplication and addition in an expression, which is why we compute powers like \(2^3\) and \(3^3\) before proceeding to other operations.

Subtraction

Subtraction is a fundamental arithmetic operation that represents the process of taking away one quantity from another. It is commonly used to calculate the difference between numbers or values. Here are some key points to keep in mind:

• The number from which you are subtracting is called the minuend.
• The number you are subtracting is called the subtrahend.
• The result is known as the difference.

In the expression \(8 - 27\), \(8\) is the minuend, \(27\) is the subtrahend, and \(-19\) is the difference.
When dealing with subtraction in algebraic expressions, it’s essential to apply the subtraction operation after handling any exponentiation or multiplication that appears earlier due to the order of operations.

Arithmetic Operations

Arithmetic operations are the basic operations used in mathematics to manipulate numbers. These operations include addition, subtraction, multiplication, and division. A full understanding of these operations is crucial for simplifying expressions and solving equations.
Here is a quick overview:

• Addition: Combines two numbers into a sum.
• Subtraction: Determines the difference between numbers.
• Multiplication: Calculates the product of numbers.
• Division: Determines how many times one number is contained within another.

In the realm of simplifying numerical expressions, it is vital to perform these operations in the correct order. This includes completing any exponentiation before performing any base arithmetic operations, such as subtraction. Once the powers are calculated as in \(2^3 = 8\) and \(3^3 = 27\), the arithmetic operation of subtraction can be correctly applied to reach the simplified result of \(8 - 27 = -19\).