Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. \(5[-1(7+12-16)+4]+2\)

Step-by-Step Solution

Answer

The simplified expression is 7.

Step 1: Simplify Inside the Parentheses

First, simplify the expression inside the innermost parentheses: \(7 + 12 - 16 = 3\). The expression now becomes: \(5[-1(3) + 4] + 2\).

Step 2: Distribute the Negative Sign Inside the Brackets

Next, distribute the \(-1\) across the expression inside the brackets: \(-1(3) = -3\).The expression now becomes:\(5[-3 + 4] + 2\).

Step 3: Simplify the Bracket

Simplify the expression inside the bracket: \(-3 + 4 = 1\).The expression now becomes:\(5[1] + 2\).

Step 4: Remove the Bracket

Since there is only a single term inside the brackets, they can be removed: \(5[1] = 5 \).The expression now simplifies to:\(5 + 2\).

Step 5: Simplify the Final Expression

Finally, add the remaining terms:\(5 + 2 = 7\).This is the simplified form of the original expression.

Key Concepts

Parentheses in Algebra

In algebra, parentheses are used to group numbers and expressions, indicating operations that should be performed first according to the order of operations. By carefully evaluating expressions inside parentheses, you ensure that calculations are done correctly.
For instance, in the expression \(5[-1(7+12-16)+4] + 2\), notice how the innermost expression is \((7+12-16)\). Calculate this part first to simplify your work and prevent errors.

• Always start simplifying from the innermost set of parentheses.
• Perform all arithmetic inside each pair of parentheses before moving outward.

Parentheses not only direct which operations to tackle first, but also help prevent misunderstandings in mathematical expressions by clarifying which numbers or terms should be multiplied, added, or subtracted together.

Distributive Property

The distributive property in algebra allows you to multiply a number by a group of numbers inside parentheses. This is a powerful tool for simplifying expressions and helps in transitioning terms correctly. The property is defined generally as \(a(b + c) = ab + ac\).
In our exercise, we see this when distributing \(-1\) across the expression inside the brackets, from \(-1(3)\) to achieving \(-3\). This means we multiply \(-1\) by each term inside the parentheses.

• It is particularly useful for eliminating parentheses and simplifying the overall expression.
• Helps in breaking down complex equations into simpler parts.

Understanding the distributive property is crucial because it saves time and simplifies processes in algebra, particularly when dealing with variables and larger sets of numbers.

Arithmetic Simplification

Arithmetic simplification involves reducing expressions to their simplest form. This makes the expression easier to handle and interpret. In our example, after distributing and simplifying the expression inside the brackets, we ended with the expression \(5 + 2\), which simplifies to \(7\) finally.

• Combine like terms to reduce expressions to their simplest form.
• Perform operations such as addition, subtraction, multiplication, or division wherever it helps to simplify.

Simplifying might also entail looking at fractions, but in this particular exercise, our focus remained on whole numbers and understanding basic operations. Mathematical expressions are more manageable and less error-prone when fully simplified.