Problem 74 - Prealgebra

Question

Simplify the given expression. \(-4.06-(4.4-(-10.04))\)

Step-by-Step Solution

Answer

The expression simplifies to -18.5.

Step 1: Simplify the Inner Parentheses

Inside the parentheses, there is an expression: \(4.4 - (-10.04)\). Subtracting a negative number is the same as adding its positive equivalent. Therefore, \(4.4 - (-10.04)\) simplifies to \(4.4 + 10.04\).

Step 2: Perform Addition

Now, add the two numbers: \(4.4 + 10.04 = 14.44\). So, the expression inside the parentheses simplifies to \(14.44\).

Step 3: Final Simplification

Substitute the expression from Step 2 back into the original equation to get \(-4.06 - 14.44\). Now, perform the subtraction: \(-4.06 - 14.44 = -18.5\).

Key Concepts

Simplifying Expressions

Simplifying expressions is like cleaning up a messy room. By reducing expressions to their simplest form, we make them easier to understand and work with.
To simplify an expression, we aim to combine like terms and perform operations in a sequence that follows mathematical rules. Here are a few tips for simplifying any expression:

• Identify Like Terms: Like terms are terms that have the same variable raised to the same power. For example, combine coefficients of terms like \(x\), \(y^2\), etc.
• Use Proper Order of Operations: Remember the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This guides us in solving expressions correctly.
• Simplify Step by Step: Take smaller sections of the expression to simplify, before putting it all together.

In the exercise provided, we simplified part of the expression step by step starting from the innermost operation.

Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign. A negative number shows a decrease or a deficit, like owing money. Understanding how to work with negative numbers is essential for simplifying expressions.
When dealing with negative numbers:

• Subtracting Negative Numbers: Subtracting a negative is the same as adding a positive. So, \(a - (-b)\) becomes \(a + b\).
• Negative and Positive Numbers: When adding and subtracting, think of negatives as going backward or downward. For example, \(-4 + 2\) is like stepping 4 steps backward and 2 forward, landing at \(-2\).
• Performing Operations: When multiplying or dividing, two negatives make a positive (\(-a)\times(-b) = ab\), but a negative and a positive make a negative (\(-a)\times b = -ab\).

In the provided exercise, we used these rules to handle the subtraction of a negative number efficiently.

Parentheses in Math

Parentheses in math are used to group parts of an expression. They help dictate the order in which operations should be performed. Think of parentheses as a spotlight highlighting what should be solved first.
To work with parentheses correctly:

• Evaluate Inside First: Always simplify the expression inside parentheses before dealing with what's outside, as in \( (4 - 2) + 3 \), which becomes \((2) + 3\).
• Nested Parentheses: If there are multiple layers, start with the most inner sets. For example, in \[ (3 - (2 +1)) + 4 \], solve \(2 + 1\) inside first.
• Distributing: When a value or term is outside parentheses, you may need to "distribute" or multiply across what’s inside, like in \(2(3 + 4) = 2\times3 + 2\times4 \).

In our exercise, we simplified an expression within parentheses by carefully following these rules.