Problem 88 - Prealgebra

Question

Given \(\mathrm{a}=-8.3, \mathrm{~b}=8.2\), and \(\mathrm{c}=5.4\), evaluate the expression \(\mathrm{ab}-\mathrm{c}^{2}\).

Step-by-Step Solution

Answer

The value of the expression is \(-97.22\).

Step 1: Calculate Product of a and b

First, calculate the product of \(a\) and \(b\) using the given values, \(a = -8.3\) and \(b = 8.2\). Thus, \(ab = (-8.3) imes 8.2\). The result is \(-68.06\).

Step 2: Calculate Square of c

Next, calculate the square of \(c\), given \(c = 5.4\). Therefore, \(c^2 = (5.4)^2 = 29.16\).

Step 3: Evaluate Expression ab - c^2

Finally, substitute the values of \(ab\) and \(c^2\) calculated in Steps 1 and 2 into the expression \(ab - c^2\): \(-68.06 - 29.16\). This gives us \(-97.22\).

Key Concepts

Evaluating Expressions

Evaluating expressions in math might seem tricky at first, but it boils down to following simple steps and rules. In this problem, you're given values for variables and need to find a number that results from doing several operations on these numbers. Here's how you can go about it easily:

• Start by identifying the components of the expression you need to evaluate. Here, that's the product of \(a\) and \(b\), and the square of \(c\).
• Proceed by substituting each variable with its given value.
• Next, perform each mathematical operation one step at a time as laid out. This might involve multiplication, addition, subtraction, or powers.
• Finally, simplify the expression by combining the results to get your final answer.

By methodically going through each step, you minimize mistakes and make sure you're executing all mathematical operations correctly.

Multiplication and Squaring

Multiplication and squaring are foundational operations in mathematics that you encounter frequently. To multiply two numbers, like \(a = -8.3\) and \(b = 8.2\), simply multiply them together to find their product. Here’s how:

• Consider the signs: A negative times a positive result in a negative. This is why \((-8.3) \times 8.2\) equals \(-68.06\).
• Multiply the absolute values and attach the proper sign.

When asked to square a number, such as \(c = 5.4\), you multiply the number by itself:

• Express the operation as \(c^2 = (5.4)^2\).
• Calculate this to get \(29.16\).

Squaring always produces a positive number, even if the original number is negative, since two negatives multiply to make a positive. These operations are crucial for solving many prealgebra problems.

Negative Numbers Operations

Working with negative numbers can be a bit confusing, but by keeping a few simple rules in mind, you can navigate these problems smoothly. When performing operations involving negative numbers:

• Multiplication & Division: A negative times a positive, or divided by a positive, always results in a negative. Like our example: \((-8.3) \times 8.2 = -68.06\).
• Addition & Subtraction: Adding a negative number is like subtracting its absolute value, while subtracting a negative is like adding its absolute value. For instance, \(-68.06 - 29.16 = -97.22\) means you’re combining two negative values.
• Squares: When you square any number, the result is always positive because you're multiplying it by itself, eliminating any negative sign. However, note that the negative sign isn't used in squaring operations unless it’s affecting the entire operation outside the parenthesis.

These rules help you understand how negative numbers behave in different scenarios, making computations straightforward and error-free.