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Chapter 1: Problem 72

Evaluate each expression for \(a=-1, b=3,\) and \(c=-4\) $$(a-c)(a+c)$$

Short Answer

Expert verified
\(-15\)

Step by step solution

01

Substitute the values

Start by substituting the given values for each variable in the expression. Given values are: \(a = -1, b = 3, c = -4\). The expression to evaluate is \((a - c)(a + c)\). Substituting the given values: \(((-1) - (-4))((-1) + (-4))\).
02

Simplify inside the parentheses

First, simplify inside the parentheses: \((-1) - (-4)\) becomes \(-1 + 4\) which equals 3. Next, simplify the other set of parentheses: \((-1) + (-4)\) becomes \(-1 - 4\) which equals -5. Therefore, the expression is now: \((3)(-5)\).
03

Multiply the results

Now, multiply the results from the simplified parentheses: \(3 \times (-5) = -15\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution
Substitution is the process of replacing variables with their given values. This is the first step when evaluating any algebraic expression. In the given exercise, you start with the expression \( (a - c)(a + c) \) and need to substitute the values \( a = -1 \), \( b = 3 \), and \( c = -4 \).
Using substitution, the expression \( (a - c)(a + c) \) becomes \( ((-1) - (-4))((-1) + (-4)) \).
Proper substitution ensures that the expression only contains numbers, making it easier to simplify and solve.
Always remember to substitute carefully and double-check to avoid errors.
Simplifying Expressions
Simplifying expressions involves performing arithmetic operations within the parentheses first.
In our example \( ((-1) - (-4))((-1) + (-4)) \), we start by simplifying each set of parentheses:
  • First, simplify \( (-1) - (-4) \). Since subtracting a negative number is the same as adding its positive, it becomes \(-1 + 4\rightarrow 3\).
  • Next, simplify \((-1) + (-4) \). Combining these values leads to \-1 - 4 = -5\.
Thus, the expression transforms into \((3)(-5) \).
Working step-by-step like this makes complex expressions much more manageable.
Order of Operations
The order of operations is crucial in mathematics. It dictates the sequence in which you solve parts of an expression to ensure accurate results.
The most common acronym for remembering this is PEMDAS, which stands for:
  • Parentheses
  • Exponents
  • Multiplication and Division (left to right)
  • Addition and Subtraction (left to right)
For our expression, after substitution, we simplify inside the parentheses first, following the rules of PEMDAS.
After simplifying each set of parentheses to get \(3 \) and \(-5\), we then proceed to the multiplication step: \((3) \times (-5) = -15 \).
This logical sequence ensures that the operations are performed correctly, leading to a reliable result.

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