Problem 59 - Elementary Algebra

Question

Find the value of \(5(8-6)+3(5+2 \cdot 3)\).

Step-by-Step Solution

Answer

Answer: 43

Step 1: Recall the Order of Operations (PEMDAS)

In order to simplify the expression correctly, remember the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Step 2: Simplify Expressions Inside Parentheses

First, we'll find the value inside each set of parentheses. We have two sets of parentheses: \((8-6)\) and \((5+2\cdot 3)\). Simplify them like this: \((8-6) = 2\) \((5+2\cdot 3) = (5+6) = 11\)

Step 3: Perform Multiplications

Next, we'll perform the multiplications. Our expression is now: \(5(2) + 3(11)\) We'll multiply each term to get: \(10 + 33\)

Step 4: Perform Addition

At last, we'll perform the addition operation: \(10 + 33 = 43\) The value of the given arithmetic expression is \(43\).

Key Concepts

PEMDAS

When faced with a complex arithmetic expression, it's imperative to follow a strict set of rules known as PEMDAS to find the correct value. PEMDAS is an acronym that helps students remember the order of operations in simplifying expressions: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. This sequence ensures accurate computation.

Let's break it down:

• Parentheses: Solve expressions inside the parentheses first. Nested parentheses are like layers of an onion - start peeling from the outermost layer.
• Exponents: Handle any exponents, which are numbers or variables raised to a power, after parenthesis.
• Multiplication and Division: Next, look for multiplication or division operations, proceeding from left to right. They are equals in the order of operations - so the leftmost operation comes first, not necessarily all multiplications before any divisions.
• Addition and Subtraction: Lastly, carry out addition and subtraction, also from left to right. Just as with multiplication and division, treat these operations as equals and follow the left-to-right rule.

Ignoring PEMDAS can result in errors, so using this mnemonic is crucial to tackling arithmetic expressions correctly. If you think of PEMDAS as a ladder, each step must be climbed in order - no skipping or jumping allowed!

Simplifying Expressions

Simplifying expressions is a bit like cleaning up a room: you make it neat by organizing and reducing things to their simplest form. It involves condensing an expression without changing its value.

Here’s what you do:

• Combine like terms. For example, if you have multiple instances of the same variable or number, you can add them together.
• Use the distributive property to remove parentheses when necessary by multiplying each term inside the parentheses by the term outside.
• Carry out any additions or subtractions with related terms.

Simplification is not just about making an expression more manageable; it also sets the stage for further operations, such as solving for variables. Each step in the process must be completed with care to avoid missteps that could change the meaning of the original expression.

Arithmetic Expressions

An arithmetic expression is a combination of numbers, operations signs, and sometimes parentheses that stands for a single number. Imagine you have a box full of different tools - each tool (operation) has a specific job, and the way you use them (the order of operations) can dramatically change the outcome of your project.

Arithmetic expressions can be as simple as \(2 + 3\) or as complex as \(5(8-6)+3(5+2 \cdot 3)\), with multiple operations. They are the cornerstone of algebra and all higher mathematics. When you encounter an arithmetic expression, it's like following a recipe; each ingredient (number) and how it's mixed (operation) must be considered step by step to achieve the desired final product - the value of the expression.

Understanding how to handle arithmetic expressions lays the foundational skills necessary for solving more advanced problems in algebra, calculus, and beyond. It’s essential to become proficient in interpreting and simplifying them, as they're not just math problems to solve - they're a language that conveys quantities and their relationships in a precise, logical manner.