Problem 70 - Prealgebra

Question

Simplify the given expression. \((-8.88)(-9.2)-(-2.3)^{2}\)

Step-by-Step Solution

Answer

76.406

Step 1: Simplify the Multiplication

First, simplify the multiplication portion of the expression: \((-8.88) \times (-9.2)\). Since both numbers are negative, their product will be positive. Calculate:\[(-8.88) \times (-9.2) = 81.696\].

Step 2: Simplify the Exponentiation

Next, focus on the exponentiation part of the expression: \((-2.3)^{2}\). Squaring a negative number results in a positive number:\[(-2.3)^{2} = (-2.3) \times (-2.3) = 5.29\].

Step 3: Subtract the Results

Now, subtract the result from the squaring calculation from the result of the multiplication.\[81.696 - 5.29 = 76.406\].

Key Concepts

Multiplication of Negative Numbers

When dealing with negative numbers, especially in multiplication, it's crucial to understand how their signs interact. Multiplying two negative numbers results in a positive product. This might seem counterintuitive at first, but there's a straightforward reason behind it:

• Negative numbers essentially "invert" values. Therefore, multiplying by a negative flips the sign of the number you're multiplying.
• When you multiply a negative by a negative, you "flip the sign" twice, bringing you back to a positive result.

In our exercise, by calculating \[(-8.88) \times (-9.2) = 81.696\]we turn the negative signs into a positive product. Think of this process as a double negation, similar to how two "not" statements negate each other in logic. Multiplication of negative numbers is often seen as cancellation of the negatives, resulting in a positive outcome.

Exponentiation

Exponentiation involves raising a number to the power of another, essentially multiplying the number by itself a certain number of times. Understanding how exponents affect negative numbers is vital:

• When a negative number is raised to an even power, like squaring in our problem, the result is positive. This happens because multiplication occurs an even number of times, flipping the negatives to a positive.
• If a negative number were raised to an odd power, the result would remain negative.

In the problem given, squaring \[(-2.3)^{2} = (-2.3) \times (-2.3)\]gives us a positive result of 5.29. The even number of multiplications leads to the positive outcome. Think of it as pairing up the negative signs in each multiplication to effectively nullify them.

Order of Operations

The order of operations is a foundational concept in mathematics that ensures expressions are simplified consistently and correctly. Often remembered as PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

This order dictates the sequence in which mathematical operations should be performed. In our exercise, it's essential to:- Perform multiplication: dealing with \[(-8.88) \times (-9.2)\] first because it's the first operation encountered.- Move on to exponentiation: \[(-2.3)^{2}\]comes next.- Finally, conduct the subtraction of results. \[81.696 - 5.29 = 76.406\]This priority sequence ensures that we simplify the expression correctly, yielding accurate results. Without following the order of operations, mathematical expressions could easily lead to incorrect outcomes.