Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

Simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. \(-4[5(-3+12-4)+2(13-7)]\)

Step-by-Step Solution

Answer

The simplified expression is -148.

Step 1: Simplify Inside the Innermost Parentheses

First, solve the expressions inside the parentheses: 1. \(-3 + 12 - 4 = 5\)2. \(13 - 7 = 6\)

Step 2: Apply the Results to the Original Expression

Replace the simplified expressions back: \(-4[5(5) + 2(6)]\)

Step 3: Simplify Inside the Brackets

Perform the operations inside the brackets: 1. Multiply the results by their respective coefficients: - \(5 \times 5 = 25\) - \(2 \times 6 = 12\)2. Add the products: - \(25 + 12 = 37\)

Step 4: Distribute the -4

Multiply the result by \(-4\): - \(-4 \times 37 = -148\)

Step 5: Final Answer

Since there are no fractions to reduce or further parentheses to eliminate, the simplified expression is \(-148\).

Key Concepts

Order of Operations

When simplifying expressions, the order of operations helps us maintain consistency and accuracy. The famous acronym PEMDAS is commonly used to remember this order:

• **P**arentheses first
• **E**xponents (or powers, such as squared or cubed)
• **M**ultiplication and **D**ivision (from left to right)
• **A**ddition and **S**ubtraction (from left to right)

In our exercise, the order of operations guides each step. Initially, parentheses are addressed. After simplifying within those, term multiplication follows. Finally, all operations are performed sequentially, as required by this hierarchy. It ensures that each step is logically sound, leading to an accurate result.

Parentheses

Parentheses play a crucial role in organizing complex expressions and dictating the sequence of operations. In our exercise, expressions within parentheses are evaluated first. This reduces the larger problem into smaller, more manageable pieces. For instance, solving \( -3 + 12 - 4 \) gives us 5, while \( 13 - 7 \) simplifies to 6.Each expression's simplification resolves to a single number, replacing the original parentheses in the expression. Simplifying in this manner helps isolate specific parts, minimizing error opportunities. It’s similar to solving parts of a puzzle individually before fitting them together. This also makes subsequent operations, like multiplication, much easier to handle.

Distributive Property

The distributive property is a handy tool for simplifying equations, allowing for flexibility in expressions containing both multiplication and addition or subtraction. According to this property, \( a(b + c) = ab + ac \).In context, let's recall the expression \( -4[5(5) + 2(6)] \). Here, first each term inside the brackets is multiplied:

• 5 times 5 results in 25
• 2 times 6 results in 12

These results are added to yield 37. Finally, multiplying the whole by -4 results in \( -148 \). This operation demonstrates how multiplying a sum by a number is the same as multiplying each addend by the number and then summing the results. It's a strategic way to break down and ultimately resolve expressions efficiently.