Problem 74 - Prealgebra

Question

Simplify the given expression. \(-2.6-(-9.8)(9.9)^{2}\)

Step-by-Step Solution

Answer

The simplified result is 957.898.

Step 1: Evaluate the Innermost Expression

First, simplify the expression inside the parentheses. Calculate the square of 9.9: \[(9.9)^2 = 9.9 \times 9.9 = 98.01\]

Step 2: Simplify the Negation

Next, simplify the expression with the negation and product: \[-9.8 \times 98.01\] Multiply to get: \[-960.498\]

Step 3: Combine the Results

Finally, substitute the value obtained into the main expression and simplify: \[-2.6 - (-960.498)\] This becomes: \[-2.6 + 960.498 = 957.898\]

Key Concepts

Evaluating Expressions

Evaluating expressions in mathematics means finding the value of the expression by working through various operations. It involves performing calculations according to specific rules, ensuring the expression is simplified correctly. To evaluate expressions efficiently, follow these simple steps:

• Identify the parts of the expression you need to solve first.
• Apply the correct operations in the right order (more on this in the "Order of Operations" section).
• Simplify progressively until you reach a single, simplified result.

When given an expression like \(-2.6-(-9.8)(9.9)^{2}\), the goal is to break it down into simpler parts that are easier to solve. Start from the innermost operation, move to any exponents, and handle both negative and positive interactions carefully.

Order of Operations

Understanding the order of operations is essential to correctly simplify mathematical expressions. This order is often remembered by the acronym PEMDAS:

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

Let's apply this order to our expression \(-2.6-(-9.8)(9.9)^{2}\):
Start by tackling the part inside the parentheses: \((9.9)^2\). Next, handle the multiplication by \(-9.8\), and finally address the subtraction that becomes addition after simplifying negatives. Proper application of this order ensures you arrive at the right answer without confusion or error.

Negative Numbers

Working with negative numbers can be tricky, but once you understand the basic concepts, you'll handle them just fine! A key rule is that subtracting a negative number is the same as adding its positive equivalent. Let's see how this works.
When you encounter \(-2.6-(-960.498)\), notice how subtracting \(-960.498\) turns into addition: \(-2.6 + 960.498\). This transformation is crucial to accurately simplify the expression.
Remember:

• Subtracting a negative: Turns into addition.
• Adding a negative: Like subtracting the positive value.
• Multiplying two negatives gives a positive result.

Each rule can change how you interpret and solve expressions, so understanding them leads to accurate solutions.

Exponents

Exponents signify repeated multiplication of a number by itself. For example, in \((9.9)^2\), you multiply 9.9 by itself to get \(98.01\). Exponents are dealt with after resolving any operations inside parentheses but before multiplications or divisions outside.
Here's a quick guide on handling exponents:

• The base (the number being multiplied) appears before the exponent symbol.
• The exponent (also known as power) shows how many times the base is multiplied by itself.
• Exponents simplify larger or more complex multiplications.

In the original expression \(-2.6-(-9.8)(9.9)^{2}\), always compute \((9.9)^2\) first, following the parentheses rule. This sets the stage for subsequent operations, ensuring that each step is based on a correct, simplified value.