Problem 86 - Elementary and Intermediate Algebra

Question

Evaluate each expression, for \(x=3, y=-2,\) and \(z=-4\) See Example 10. $$ [(z-1)(z+1)]^{2} $$

Step-by-Step Solution

Answer

The value of the expression is 225.

Step 1: Substitute the Values into the Expression

Take the given expression \([(z-1)(z+1)]^2\) and substitute the values for the variables: \(z = -4\). This gives us the new expression \([(-4-1)(-4+1)]^2\).

Step 2: Simplify Inside the Parentheses

First simplify each part inside the parentheses: Calculate \(-4 - 1\) which equals \(-5\) and \(-4 + 1\) which equals \(-3\). The expression is now \([(-5)(-3)]^2\).

Step 3: Multiply the Terms Inside Parentheses

Multiply the two results: \(-5 \times -3 = 15\). The expression is now \(15^2\).

Step 4: Evaluate the Squared Term

Calculate \(15^2\), which is \(15 \times 15 = 225\).

Key Concepts

Substitution Method

The substitution method is a fundamental concept in algebra that simplifies an expression by replacing the variables with their given values. This technique is a crucial first step in solving algebraic expressions.

In our exercise, the expression is \( [(z-1)(z+1)]^{2} \) and we are given that \( z = -4 \). The substitution method involves:

• Identifying the variables in the expression, which is only \( z \) in this case.
• Rewriting the expression by plugging in the value of the variable, so \( [-4-1][-4+1] \) becomes \( [(-4-1)(-4+1)]^{2} \).

This process converts an abstract form into a concrete numerical expression ready for further simplification. By successfully substituting, you maintain the integrity of the expression to proceed with other operations.

The key to mastering substitution is consistent practice and ensuring each variable is replaced correctly without modifying the mathematical structure.

Simplifying Expressions

Simplifying expressions is like untangling a knot—it makes solving the equation easier. This process involves breaking down the expression step by step.

Once substitution is done, simplify each section inside the expression. For \( [(-4-1)(-4+1)]^{2} \):

• Simplify each parenthesis first: \( -4 - 1 \) results in \( -5 \), while \( -4 + 1 \) results in \( -3 \).
• Multiply the simplified results from the parentheses: \( (-5) \times (-3) = 15 \).

Simplifying at this stage reduces the bulk of the expression, changing it into \( 15^{2} \), thus making it more straightforward to solve. This method ensures each arithmetic operation follows from the correct simplification of previous terms.

Simplifying expressions not only helps in managing complexity but also in understanding the relationships between numbers and operations.

Order of Operations

The order of operations is a set of rules followed to ensure that expressions are solved correctly and universally. Remember the helpful acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition, and Subtraction (from left to right).

In our expression \( 15^{2} \), after substitution and simplification, we deal with the exponent:

• Calculate the exponent \( 15^{2} \), which means \( 15 \times 15 \). This results in \( 225 \).

Following the order of operations is crucial when dealing with more complex calculations involving multiple layers of arithmetic to ensure accuracy and consistency.

By adhering to these rules, you maintain clarity and correctness in solving any mathematical expression, just as we did with this exercise.