Problem 83 - Prealgebra

Question

Given \(a=-2.4, b=-2.1\), and \(c=-4.6\), evaluate the expression \(a b-c^{2}\)

Step-by-Step Solution

Answer

The value of the expression is -16.12.

Step 1: Understand the Expression

We are given the expression \( ab - c^2 \) with specific values for variables: \( a = -2.4 \), \( b = -2.1 \), and \( c = -4.6 \). Our task is to substitute these values into the expression and calculate the result.

Step 2: Substitute the Values

Replace \( a \), \( b \), and \( c \) in the expression \( ab - c^2 \) with \( -2.4 \), \( -2.1 \), and \( -4.6 \) respectively. This gives us:\[ab - c^2\ = (-2.4)(-2.1) - (-4.6)^2.\]

Step 3: Calculate the Product \( ab \)

Calculate \( ab = (-2.4) \times (-2.1) \). Since both numbers are negative, the product is positive:\[ab = 2.4 \times 2.1 = 5.04.\]

Step 4: Calculate the Square \( c^2 \)

Calculate \( c^2 = (-4.6)^2 \). Squaring a negative number gives a positive result:\[c^2 = 4.6 \times 4.6 = 21.16.\]

Step 5: Final Calculation

Plug the calculated values back into the expression \( ab - c^2 \):\[ab - c^2 = 5.04 - 21.16.\]Subtract to find the final value:\[5.04 - 21.16 = -16.12.\]

Key Concepts

Substitution Method

The substitution method is widely used in algebra to simplify expressions and solve equations. It involves replacing variables in an expression with specific values. For example, in this exercise, we were given the expression \(ab - c^2\) and specific values for \(a = -2.4\), \(b = -2.1\), and \(c = -4.6\). By substituting these values into the expression, we transform abstract components into numbers that can be computed.

• First, locate each variable in the expression.
• Replace each variable with its given value.
• Rewrite the expression with the substituted values before performing any calculations.

This method simplifies the process of working with complex algebraic expressions, making it easier to focus on arithmetic operations. After substitution, the expression \(ab - c^2\) becomes \((-2.4)(-2.1) - (-4.6)^2\).

Arithmetic Operations

Arithmetic operations are basic mathematical procedures used to perform calculations. These include addition, subtraction, multiplication, and division. In this exercise, we particularly use multiplication and subtraction.Multiplication: We start by finding the product \(ab\). Multiplying two negative numbers, like \((-2.4)\) and \((-2.1)\), results in a positive outcome because the negatives cancel. Thus, \((-2.4)\times(-2.1) = 5.04\).Squared Operation: Squaring a number entails multiplying it by itself. In the expression \(c^2\), we square \((-4.6)\) resulting in a positive \(21.16\).Subtraction: Finally, the subtraction operation brings us to the last step, combining the results: \(5.04 - 21.16\). Since 21.16 is larger than 5.04, the result is negative, \(-16.12\).The use of these operations forms the backbone of solving algebraic expressions.

Negative Numbers

Understanding negative numbers is crucial when dealing with algebraic expressions and arithmetic operations. Negative numbers are those that are less than zero, represented with a minus sign (-) before them. In calculations, negative numbers follow specific rules:

• When you multiply two negative numbers, the result is always positive, as their effects cancel each other out. Thus, \((-2.4)\times(-2.1) = 5.04\).
• Squaring a negative number also results in a positive number, because two negatives make a positive. For example, \((-4.6)^2 = 21.16\).
• Subtracting a larger number from a smaller one, as \(5.04 - 21.16\), results in a negative result: \(-16.12\).

By understanding these rules, you can accurately solve problems involving negative numbers without confusion. Remember, negative signs significantly impact the outcome of calculations.