Problem 82 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression by removing parentheses and brackets. $$4[6(x-3)+1]$$

Step-by-Step Solution

Answer

The simplified form of the algebraic expression is \(24x - 68\).

Step 1: Applying the distributive property inside the brackets

Take 6 (inside the brackets) and distribute it to the terms inside the parenthesis by multiplying. This results in \(4[6x - 18 + 1]\). Now add the numbers inside the brackets which gives \(4[6x - 17]\).

Step 2: Applying the distributive property to the brackets

Now distribute the 4 to the terms inside the brackets by multiplying. The final simplified algebraic expression is \(24x - 68\).

Key Concepts

Distributive Property

The distributive property is a very helpful mathematical rule that lets you simplify expressions much more easily. It is used when you multiply a number by a sum (or difference) inside parentheses or brackets. This property means you need to "distribute" the multiplication over each term inside. So, if you have an expression like \(a(b + c)\), the distributive property lets you rewrite it as \(ab + ac\).

In the exercise above, the distributive property was first applied within the parentheses and then within the brackets.
Here's how it worked:

• First, the number 6 was multiplied by both terms \((x - 3)\) inside the parentheses. This meant doing \(6 \times x\) and \(6 \times -3\), giving \(6x - 18\).
• Next, the number 4 was distributed to each of the terms left inside the brackets \([6x - 17]\). That involved doing \(4 \times 6x\) and then \(4 \times -17\), resulting in the final simplified expression \(24x - 68\).

Without the distributive property, simplifying complex algebraic expressions would be much more difficult.

Simplification

Simplification in algebra is about making expressions less complicated while still keeping them equivalent. It's like cleaning up and organizing a cluttered space! The goal is to express the algebraic statement in the simplest form possible.

When simplifying, you should:

• Perform all possible arithmetic operations.
• Combine like terms (terms with the same variable and power).
• Remove any unnecessary parentheses or brackets.

In this particular example, simplification was achieved by applying the distributive property effectively. By taking away the parentheses and reducing the arithmetic within, you arrive at a neat expression. This process of breaking down the expression while keeping all terms in their rightful place is crucial in algebra. It helps you better understand the relationship between variables and constants in an equation.

Brackets and Parentheses

Brackets and parentheses in algebraic expressions signal which operations should be performed first. It's like following a to-do list; they prioritize the steps needed to solve the math problem.

Here are some tips on using brackets and parentheses:

• Operations inside the parentheses are completed first, adhering to the order of operations (often remembered by PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication, Division, Addition, and Subtraction).
• If there are brackets outside parentheses, resolve what's inside the parentheses before moving to the brackets.

This exercise demanded understanding what to do inside the parentheses \((x - 3)\) first before moving on to the brackets \([6(x - 3) + 1]\). After simplification within parentheses, you manage the operation inside the brackets by applying the distributive property and combining like terms.

Remember, paying attention to brackets and parentheses ensures you're conducting operations in the right order and simplifies complex expressions step-by-step.