Problem 43 - Prealgebra

Question

Evaluate each expression. $$2+3 \cdot 5$$

Step-by-Step Solution

Answer

The expression evaluates to 17.

Step 1: Understand the Order of Operations

In order to evaluate the expression, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). In this expression, we have addition and multiplication.

Step 2: Identify the Operations

The expression given is \(2 + 3 \cdot 5\). There are two operations present: multiplication (\(3 \cdot 5\)) and addition (\(2 + \)). According to PEMDAS, we perform multiplication before addition.

Step 3: Perform the Multiplication

First, solve the multiplication part of the expression: \(3 \cdot 5 = 15\).

Step 4: Perform the Addition

Next, add the result of the multiplication to 2: \(2 + 15 = 17\).

Step 5: Final Answer

There are no more operations left to perform. The final result of the expression is \(17\).

Key Concepts

PEMDAS

The acronym **PEMDAS** is essential when evaluating mathematical expressions. It stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This sequence dictates the operation order to solve expressions accurately.

• **Parentheses**: Begin with calculations inside parentheses.
• **Exponents**: Next, calculate powers and roots.
• **Multiplication and Division**: Proceed left to right as they appear in the expression.
• **Addition and Subtraction**: Complete these last, also moving from left to right.

Understanding PEMDAS ensures that each part of an expression gets solved in the correct order, preventing errors. For instance, in the expression \(2 + 3 \cdot 5\), PEMDAS tells us to multiply first, then add.

Arithmetic Expressions

When you see an **arithmetic expression**, it consists of numbers and operators like addition, subtraction, multiplication, and division. Evaluating an arithmetic expression involves simplifying it step by step to find its value.

In our example, the expression is \(2 + 3 \cdot 5\). It combines the operations of multiplication and addition. When simplifying, always pay attention to the operation precedence provided by PEMDAS.

Simplification transforms a complex expression into a single value by evaluating the terms according to set rules—ensuring coherence and correctness in math tasks. Each step in simplifying an arithmetic expression should follow logical rules, and a misunderstanding in the order can lead to errors. Take each operator one at a time, and don't rush.

Multiplication and Addition

Two fundamental operations in mathematics are **multiplication and addition**. They are used frequently to evaluate and solve various expressions.

• **Multiplication** forms when a number is added to itself a certain number of times, like \(3 \cdot 5\), which means adding 3 five times, leading to 15.
• **Addition** involves counting things together. After multiplication, adding numbers is the next step, such as adding 2 to the product of 3 and 5. Therefore, \(2 + 15 = 17\).

When dealing with both in an expression, multiplication should come first, followed by addition, unless parentheses dictate otherwise. Correctly applying these operations ensures accurate results in mathematical tasks.