Problem 47 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. (Lessons 1.2,1.3) $$ 5^{3}+12 $$

Step-by-Step Solution

Answer

The result of the expression \(5^{3}+12\) is 137.

Step 1: Calculating \(5^{3}\)

The expression \(5^{3}\) means '5 multiplied by itself 3 times', which is the same as \(5 * 5 * 5\). When multiplying this out, \(5 * 5 = 25\), and then \(25 * 5 = 125\). So, \(5^{3}\) equals 125.

Step 2: Adding 12

The next step is to add 12 to the result from Step 1. This means, we'll add 12 to 125, which equals 137.

Key Concepts

Order of Operations

When evaluating mathematical expressions, we use the **Order of Operations** to determine which arithmetic operations to perform first. This is crucial for achieving consistent and correct results.
The standard order can be remembered using the acronym PEMDAS:

• **P**arentheses: Complete any calculations inside parentheses first.
• **E**xponents: Next, evaluate exponents (or powers).
• **M/D**: Multiplication and Division follow, from left to right. These two operations are on the same level and should be handled in the order they appear.
• **A/S**: Lastly, perform Addition and Subtraction, also from left to right. Similar to multiplication and division, addition and subtraction are of equal precedence.

In the exercise at hand, we dealt with the expression \(5^{3} + 12\). According to PEMDAS, we first calculated the exponent, \(5^{3}\), before proceeding to the addition of 12.

Arithmetic Operations

Arithmetic operations are the basic building blocks of mathematics, allowing us to perform fundamental calculations. There are four primary operations:

• **Addition (+)**: Combines two numbers to get a sum.
• **Subtraction (-)**: Calculates the difference between two numbers.
• **Multiplication (×)**: Computes the product of two numbers.
• **Division (÷)**: Splits one number into a specific number of parts.

In the original exercise, we encountered **exponentiation**, which is a form of repeated multiplication. Specifically, \(5^{3}\) is read as "5 raised to the power of 3," or "5 multiplied by itself 3 times." After evaluating the exponent, we performed **addition** by adding 12 to the result from the previous step. Understanding these operations is key to solving any arithmetic problem.

Mathematical Expressions

A mathematical expression is a combination of numbers, operators, and sometimes variables that represents a specific value. Expressions can vary from simple to complex, but they always consist of terms connected by arithmetic operations.
In the example \(5^{3} + 12\), we have a simple expression consisting of two terms: the power \(5^{3}\) and the number 12.
Expressions do not include an equals sign; therefore, they do not form a complete equation, like an equality would. When evaluating expressions, we perform calculations to find the value that the expression represents.

• To evaluate an expression, follow the order of operations.
• Simplify the terms by performing any calculations necessary.
• If the expression includes variables, substitute the known values before calculating.

Mastering the evaluation of mathematical expressions lays a foundation for more advanced math concepts.