Problem 65 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ 7[3(6-2)]-64 $$

Step-by-Step Solution

Answer

The simplified expression is 20.

Step 1: Simplify the Expression Inside the Parentheses

First, let's simplify the expression inside the innermost parentheses: \( 6 - 2 \). Subtract 2 from 6:\[6 - 2 = 4\]Replacing \(6 - 2\) with 4 in the expression, the expression becomes:\[7[3(4)] - 64\]

Step 2: Multiply Numbers Inside the Brackets

We now simplify the expression inside the brackets: \(3 \times 4\).Multiply 3 by 4:\[3 \times 4 = 12\]Substitute the result back into the expression:\[7[12] - 64\]

Step 3: Multiply by the Number Outside the Brackets

Now multiply the number outside the brackets (7) by the number inside the brackets (12):\[7 \times 12 = 84\]Replace \(7[12]\) with 84 in the expression, leading to:\[84 - 64\]

Step 4: Subtract the Appropriate Numbers

Finally, subtract 64 from 84:\[84 - 64 = 20\]The expression simplifies to 20.

Key Concepts

Order of Operations

Understanding the order of operations is essential to simplifying expressions correctly. Imagine trying to bake a cake by doing the steps in random order—it wouldn't work well! Math works the same way. We use a specific set of rules to decide the sequence in which parts of a math problem are solved.

• The Order of Operations can be remembered using the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• The order is crucial because mathematical operations can change the outcome depending on the sequence used.

First, focus on any operations inside parentheses, as these are prioritized. Next, handle exponents. Then, proceed with multiplication and division in the sequence they appear from left to right. Finally, wrap it up with addition and subtraction, also from left to right.
In our exercise, we first addressed the parentheses to ensure we followed this correct sequence, leading to a precise solution every time.

Parentheses in Math

Parentheses in math act as a signal to perform the operations within them first, overriding the usual order. They can be a little like traffic lights, telling you to "stop" and solve here before moving on.

• Operations within parentheses are calculated before any other outside or alongside operations.
• It's important to solve expressions inside the "innermost" parentheses first if there are multiple layers.

In our example, the expression inside the parentheses was "6 - 2". By treating this as our first priority, we ensure that we simplify correctly according to the rules.
Once simplified, this allows us to move on to the next operations, confident that all components inside those parentheses are in their simplest form.

Arithmetic Operations

Arithmetic operations are the basic building blocks of math and involve addition, subtraction, multiplication, and division. Each operation has specific roles and interactions with numbers.
In our exercise, various arithmetic operations are used in sequence:

• Subtraction: We start by subtracting 2 from 6 inside the parentheses.
• Multiplication: Then, 3 is multiplied by the result of the subtraction.
• Once more,! Multiplication appears when 7 multiplies the outcome of the brackets.
• Finally, subtraction: To complete the expression, we subtract 64 from the previously arrived number.

Each step involves applying basic arithmetic rules to simplify and solve the expression efficiently. Recognizing these patterns in arithmetic operations helps not only in this exercise but in myriad mathematical problems you may face.