Problem 21 - Intermediate Algebra

Question

In the following exercises, simplify each expression. $$ 2+8(6+1) $$

Step-by-Step Solution

Answer

The simplified expression is 58.

Step 1: Identify the Expression

Start by identifying the expression that needs to be simplified: \(2+8(6+1)\).

Step 2: Simplify Inside the Parentheses

First, simplify the expression inside the parentheses: \(6+1=7\). The expression now becomes \(2+8 \times 7\).

Step 3: Perform Multiplication

Next, perform the multiplication: \(8 \times 7=56\). Now the expression is \(2+56\).

Step 4: Perform Addition

Finally, add the remaining numbers: \(2+56=58\).

Key Concepts

Order of Operations

One of the most important concepts in algebra is the order of operations. This is a set of rules that ensures calculations are performed in the correct order. Without these rules, the same expression could be simplified in different ways by different people, leading to inconsistent results.

• First, perform any calculations inside parentheses. This ensures that grouped numbers are dealt with first.
• Next, handle exponents if there are any. These come just after parentheses.
• Multiplication and division are done next, from left to right.
• Finally, perform any addition and subtraction, again from left to right.

Remembering this order is crucial to simplifying algebraic expressions correctly. An easy way to recall this is the acronym PEMDAS - Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Parentheses

Parentheses are used in algebra to group parts of an expression that should be evaluated first. For example, in the expression \(2 + 8(6+1)\), the part within the parentheses, \(6 + 1\), is calculated before anything else.
By simplifying inside the parentheses, you reduce the expression to a simpler form. Here:

• Calculate \(6 + 1\)
• This results in \(2 + 8 \times 7\)

Evaluating what’s inside the parentheses first helps to avoid mistakes and ensures consistent results across different problems.

Multiplication in Algebra

Multiplication is another fundamental operation in algebra. After dealing with parentheses, you generally perform any multiplications next. In our example, after simplifying inside the parentheses, you get \(2 + 8 \times 7\). Now, the next step is to handle the multiplication.

• Multiply 8 by 7 to get 56
• Thus, \(2 + 8 \times 7\) simplifies to \(2 + 56\)

Dealing with multiplications before additions and subtractions ensures that the mathematical relationships in the expression are preserved.

Addition in Algebra

Finally, addition is typically performed once all other operations have been accounted for in the order of operations. After taking care of multiplication, in the expression \(2 + 56\), the next step is to perform the addition.

• Simply add 2 and 56 to get 58

Bringing it all together, our original expression \(2 + 8(6+1)\) simplifies down step-by-step to 58. Understanding how to handle addition following other operations like multiplication and parentheses is essential for simplifying algebraic expressions correctly and consistently.