Problem 112 - Intermediate Algebra

Question

In the following exercises, simplify each expression. $$ 11-3[7-4(-2)] $$

Step-by-Step Solution

Answer

First, simplify inside the parentheses: .

Step 1: Apply algebraic rules

Use properties of exponents, radicals, fractions, or algebraic identities to simplify the expression.

Step 2: State the result

The simplified expression is First, simplify inside the parentheses: ..

Key Concepts

Order of Operations

When simplifying algebraic expressions, it's essential to follow the order of operations. This ensures every calculation is done correctly. The order of operations can be remembered using the acronym PEMDAS:

• **P**arentheses
• **E**xponents
• **M**ultiplication
• **D**ivision
• **A**ddition
• **S**ubtraction

Always start with calculations inside the parentheses. After that, take care of any exponents. Then, move from left to right, handling multiplication and division. Finally, continue left to right with addition and subtraction.

Distributive Property

The distributive property is a key algebraic tool. It allows you to multiply a single term by each term within parentheses. Here's the formula for the distributive property: [ a(b + c) = ab + ac ] Essentially, you distribute the term outside the parentheses to each term inside individually.
For example, with the expression [ 3(4 - 2) ], you distribute the 3 to get [ 3*4 - 3*2. ] This results in [ 12 - 6 = 6. ] Applying the distributive property helps in simplifying complicated expressions.

Parentheses in Algebra

Parentheses play a crucial role in algebra. They dictate which operations to perform first within an expression. Without parentheses, calculations would become confusing, and the order of operations might be unclear.
Consider the expression [ 11 - 3[7 - 4(-2)] ]. Start by simplifying the innermost parentheses: [ 4(-2) = -8 ]. Then, proceed to the next set of parentheses: [ 7 - (-8) = 7 + 8 = 15. ] Finally, handle the outermost calculations: [ 11 - 3*15 = 11 - 45 = -34. ] Using parentheses helps in breaking down the problem into manageable steps. It provides clarity and ensures correct simplification.