Problem 87 - Prealgebra

Question

Given \(\mathrm{a}=-4.5, \mathrm{~b}=-6.9\), and \(\mathrm{c}=4.6\), evaluate the expression \(\mathrm{ab}-\mathrm{c}^{2}\)

Step-by-Step Solution

Answer

The expression evaluates to 9.89.

Step 1: Understand the Expression

We are given the expression \(ab - c^2\), where \(a = -4.5\), \(b = -6.9\), and \(c = 4.6\). Our task is to evaluate this expression by substituting the given values for \(a\), \(b\), and \(c\).

Step 2: Calculate \(ab\)

To find \(ab\), multiply \(a\) and \(b\). Substitute the values: \(ab = (-4.5) \times (-6.9)\). The product of two negative numbers is positive, so compute:\[ab = 4.5 \times 6.9 = 31.05\]

Step 3: Calculate \(c^2\)

This step involves finding \(c^2\). Substitute the value of \(c\):\[c^2 = (4.6)^2 = 21.16\]

Step 4: Substitute and Evaluate the Expression

Put the values of \(ab\) and \(c^2\) into the expression \(ab - c^2\):\[ab - c^2 = 31.05 - 21.16 = 9.89\]

Step 5: Conclusion

The expression evaluates to \(9.89\).

Key Concepts

Evaluating Expressions

Evaluating expressions is a fundamental skill in algebra. It involves plugging in values for variables and performing arithmetic operations to find the result. In the expression \(ab - c^2\), we start by identifying the variables: \(a\), \(b\), and \(c\). We need to replace these variables with their given values.

• \(a = -4.5\)
• \(b = -6.9\)
• \(c = 4.6\)

Once the values are substituted into the expression, our work is to perform the multiplication and subtraction as indicated by the expression. By evaluating these expressions correctly, we make abstract mathematical concepts more tangible.

Multiplication of Negative Numbers

One intriguing aspect of mathematics is understanding how negative numbers behave, especially during multiplication. When both numbers are negative, their product is positive. This might seem counter-intuitive at first, but it makes sense when you think about it as taking a negative direction a negative number of times, which essentially reverses the direction.
Using our example from the expression \(ab = (-4.5) \times (-6.9)\), note that:

• Both -4.5 and -6.9 are negative
• Their multiplication yields a positive result: \(31.05\)

Understanding this rule will simplify your approach to equations and ensure you avoid mistakes when working with negatives.

Square of a Number

Squaring a number refers to multiplying the number by itself. When calculating squares, it's important to correctly multiply the number twice. In our expression, this means taking \(c = 4.6\) and computing \(4.6 \times 4.6\).
Here's how it looks:

• \(c^2 = (4.6)^2 = 21.16\)

Squaring can be a straightforward operation, but accuracy is key. Using a calculator ensures that you compute the square precisely. Mastering this fundamental operation is crucial for higher-level math, where squaring frequently appears in formulas and equations.