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Chapter 2: Problem 69

Find the value of each expression for the given values. \(n^{2}-m^{2} ; n=-3\) and \(m=-8\)

Short Answer

Expert verified
The value is -55.

Step by step solution

01

Substitute the Values

Start with the expression \(n^2 - m^2\). Substitute \(n = -3\) and \(m = -8\) into the expression, resulting in \((-3)^2 - (-8)^2\).
02

Calculate Squares

Calculate the square of each substituted value: \((-3)^2 = 9\) and \((-8)^2 = 64\). This gives us the expression \(9 - 64\).
03

Perform the Subtraction

Subtract the second term from the first: \(9 - 64 = -55\).

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Key Concepts

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

Expression Evaluation
Expression evaluation in algebra involves calculating the value of an algebraic expression. To evaluate an expression, you substitute numbers in place of variables and then perform the necessary operations. In this example, the expression given was \(n^2 - m^2\). By evaluating it with provided values (\(n = -3\) and \(m = -8\)), we follow a sequence of operations including squaring numbers and performing subtraction.
  • First, substitute the given numbers for the respective variables.
  • Second, carry out the mathematical operations involved in the expression.
Breaking down the expression evaluation into these steps makes it straightforward and manageable.
Substitution Method
The substitution method is crucial for evaluating expressions. It involves replacing a variable with a specific value. For instance, in the expression \(n^2 - m^2\), we substitute \(n\) with \(-3\) and \(m\) with \(-8\). This substitution transforms the original algebraic expression into a numerical one.
  • Identify the variables in the expression.
  • Replace each variable with the provided value.
  • The resulting expression is purely numerical and needs no further variable manipulation.
Using this method helps simplify complex expressions and makes calculation easier. This approach is like plugging in numbers for letters, making the math problem more tangible and solvable.
Squares of Numbers
Squaring a number means multiplying the number by itself. This is an essential operation in Intermediate Algebra, especially in expressions involving exponents. It鈥檚 a two-step mechanism with straightforward but unique implications:
  • Calculate the square of negative numbers carefully, as the result is always positive. For example, \((-3)^2 = 9\) and \((-8)^2 = 64\).
  • Ensure accurate squaring before proceeding with other operations in the expression to avoid miscalculations.
Understanding and applying the concept of squares of numbers is fundamental in simplifying expressions effectively, particularly those involving quadratic forms.

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