Problem 51 - Beginning and Intermediate Algebra

Question

Simplify each expression. (Remember the order of operations.) See Examples 4 and 5. $$ 2-3(8-6) $$

Step-by-Step Solution

Answer

The simplified expression is \(-4\).

Step 1: Identify terms and operations

The expression is given as \(2 - 3(8 - 6)\). Note that the expression inside the parentheses \((8-6)\) should be calculated first according to the order of operations (PEMDAS/BODMAS).

Step 2: Simplify inside the parentheses

Calculate the operation inside the parentheses first: \(8 - 6 = 2\). The expression now reads \(2 - 3 imes 2\).

Step 3: Apply multiplication

Next, perform the multiplication: \(3 imes 2 = 6\). Substitute back into the expression to get \(2 - 6\).

Step 4: Perform subtraction

Finally, perform the subtraction: \(2 - 6 = -4\).

Key Concepts

Simplifying Expressions

In mathematics, simplifying expressions is about making a problem easier to solve. Expressions are combinations of numbers, variables, and operations. The goal of simplifying an expression is to make it shorter and more understandable.
This is done by carrying out the operations and reducing it to its simplest form. Here are some key steps to simplify expressions:

• Recognize the order of operations. This is essential to ensure calculations are performed correctly.
• Identify terms that can be combined. This involves looking for similar numbers or variables.
• Perform the necessary arithmetic operations step-by-step.

When you simplify an expression like \(2 - 3(8 - 6)\), follow these steps to break it down. Work through each part methodically to arrive at the simplest form, in this case, \(-4\).

Parentheses

Parentheses are crucial in mathematical expressions. They dictate which operations should be carried out first, according to the order of operations rule. This rule is often remembered by the acronym PEMDAS/BODMAS. The letters stand for Parentheses (Brackets), Exponents (Orders), Multiplication, Division, Addition, and Subtraction.

• Operations inside parentheses are prioritized first.
• After parentheses, any powers or exponents should be processed.
• Multiplication and division come next, working from left to right.
• Finally, perform any addition and subtraction, also from left to right.

In the example \(2 - 3(8 - 6)\), solving the operation within the parentheses \((8 - 6)\) is crucial to simplify the expression correctly. Here, it changes the equation to \(2 - 3 \times 2\) by simplifying the inside of the parentheses first.

Multiplication and Subtraction

Multiplication and subtraction are basic arithmetic operations. In expressions, they are performed in order based on the precedence rules following any operations inside parentheses. After dealing with anything in parentheses, multiplication is usually the next step.

• Identify multiplication within the expression and perform it before moving on to addition or subtraction.
• Once multiplication is handled, proceed with subtraction or addition to complete the simplifying process.

For instance, in \(2 - 3 \times 2\), after resolving the multiplication to get \(6\), you would continue with subtraction, resulting in \(-4\). Understanding this sequence helps manage more complicated expressions accurately and efficiently.