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Chapter 2: Problem 64

Evaluate each expression if \(f=-9, g=-6,\) and \(h=8\). $$-5 f g$$

Short Answer

Expert verified
The evaluated expression is -270.

Step by step solution

01

Substitute Values

Substitute the given values for the variables in the expression. The expression is given as \(-5 f g\). Replace \(f\) with \(-9\) and \(g\) with \(-6\). The expression becomes: \[-5 (-9) (-6)\]
02

Simplify Inner Multiplication

Simplify the multiplication inside the parentheses first. Multiply \(-9\) and \(-6\), which gives:\[-9 \times (-6) = 54\]
03

Multiply with Coefficient

Now multiply the result from Step 2 by \(-5\): \[-5 \times 54 = -270\]

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Key Concepts

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

Substitution
When evaluating algebraic expressions, the first step is often substitution. This involves replacing the variable in the expression with the specific value it represents. In our example, we have the expression \(-5fg\), where \(f\) is given as \(-9\) and \(g\) as \(-6\). Using substitution, you simply place the values into the expression, transforming it into \(-5(-9)(-6)\).
  • The purpose of substitution is to simplify expressions to a form where calculations can directly occur.
  • It sets the stage for further mathematical operations, like multiplication in this example.
Remember that the sign of the number must be included with its value during substitution, as it affects the result of any arithmetic operations following it.
Multiplication
Multiplication is the arithmetic operation used in our expression after substitution. We often perform multiplication in stages to simplify complex expressions. In this example, after substituting values, the expression becomes \(-5(-9)(-6)\). To systematically simplify, focus first on the multiplication of the two inner numbers:
  • Multiply \(-9\) by \(-6\), noting that the product of two negative numbers is positive. Thus, \(-9 \times (-6) = 54\).
  • Next, multiply the result by \(-5\), yielding \(-5 \times 54 = -270\).
Breaking multiplication into manageable steps helps reduce errors and makes complex computations easier to manage.
Negative Numbers
Working with negative numbers can be tricky because they have specific properties during arithmetic operations. In multiplication:
  • When two numbers with the same sign (both positive or both negative) are multiplied, the result is always positive.
  • Conversely, if the numbers have different signs (one positive and one negative), the result is negative.
In our exercise, that explains why the multiplication \(-9 \times (-6) = 54\) results in a positive number and why the subsequent multiplication by \(-5\) gives a negative result, \(-270\). Properly recognizing and applying these rules is crucial for maintaining accuracy in mathematical operations involving negative numbers.

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