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Chapter 2: Problem 86

Evaluate ab for the given replacement values. See Example 15. \(a=-8\) and \(b=8\)

Short Answer

Expert verified
The value of \( ab \) is -64.

Step by step solution

01

Understanding the Problem

We need to evaluate the expression \( ab \) given replacement values for \( a \) and \( b \). These values are \( a = -8 \) and \( b = 8 \).
02

Substitute Values into the Expression

Substitute \( a = -8 \) and \( b = 8 \) into the expression \( ab \). This gives us \( ab = (-8)(8) \).
03

Calculate the Product

Multiply the values: \(-8 \times 8 = -64\).
04

Conclusion

The value of \( ab \) after substituting the values \( a = -8 \) and \( b = 8 \) and performing the multiplication is \(-64\).

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Key Concepts

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

Variables in Algebra
Variables are fundamental components in algebra, representing unknown or changeable values in expressions and equations. In our exercise, the variables are denoted as \( a \) and \( b \). These symbols stand in for any number until we assign them specific values.

Here's why variables are so useful:
  • They allow us to generalize mathematical situations.
  • We can solve problems with different numbers without creating new equations.
  • Variables help in forming equations which can be solved for unknowns.
When we evaluate an expression like \( ab \), using specific values for \( a \) and \( b \), we transform the algebraic expression into a numerical one for which we can calculate the result. This transformation is done by substituting each variable with the given numbers.
Multiplication of Integers
When multiplying integers, the rules of signs are crucial. In our example, we needed to multiply \(-8\) by \(8\). Here's how multiplication of integers works:
  • When multiplying two positive numbers or two negative numbers, the product is positive. For instance, \(5 \times 3 = 15\) and \((-5) \times (-3) = 15\).
  • When multiplying a positive number by a negative number, the product is negative. For example, \(5 \times (-3) = -15\).
  • The same rule applies when a negative number is multiplied by a positive number, like \((-8) \times 8 = -64\).
In essence, remember that the product of two numbers with different signs is negative. This fundamental rule helps in determining the sign of the result quickly.
Evaluating Expressions
Evaluating expressions is the process of substituting variables in an expression with assigned values, and then performing the arithmetic operations to get a result. The exercise's expression \( ab \) is evaluated using the values \( a = -8 \) and \( b = 8 \).

The steps to evaluate are:
  • Substitute the values of the variables into the expression. For \( ab \), this means replacing \( a \) with \(-8\) and \( b \) with \(8\).
  • Perform the arithmetic operation of multiplication to find the result.
The operation given by substituting into the expression \( ab = (-8)(8) \), gives the result \(-64\) after multiplying. Evaluating expressions involves making sure the substitutions are correct and calculations are accurately done, ensuring the proper handling of operations such as multiplication or division.

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