Problem 56 - Algebra A Combined Function

Question

Simplify each expression. \(-5+[(4-15)-(-6)-8]\)

Step-by-Step Solution

Answer

The simplified expression is \(-18\).

Step 1: Simplify Inside the Innermost Parentheses

The expression is given as \[-5+[(4-15)-(-6)-8]\]Start by simplifying inside the innermost parentheses first. In this expression, simplify \(4 - 15\).This results in:\[4 - 15 = -11\]

Step 2: Remove Negative Signs in Parentheses

Now replace \(4-15\) with \(-11\) inside the expression:\[-5+[(-11)-(-6)-8]\]Next, remove the negative sign from \(-(-6)\) to get \(+6\).The expression becomes:\[-5+[(-11)+6-8]\]

Step 3: Simplify the Remaining Terms Inside the Brackets

Now, calculate the expression inside the brackets. Start by adding \(-11+6\):\[-11 + 6 = -5\]Next, subtract 8:\[-5 - 8 = -13\]The new expression is:\[-5 + (-13)\]

Step 4: Simplify the Expression by Removing the Brackets

Simplify the expression by removing the brackets and combing terms:\[-5 + (-13)\]This is the same as:\[-5 - 13\]

Step 5: Final Calculation

Finally, calculate \(-5 - 13\):The result is:\[-18\]

Key Concepts

Order of Operations

Understanding the order of operations is crucial to simplifying expressions correctly. In algebra, we have a specific sequence in which operations should be carried out. This sequence is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition, and Subtraction (from left to right).
When simplifying the expression \(-5+[(4-15)-(-6)-8]\), we begin by addressing the operations within the parentheses. Inside the brackets, different operations may need to be performed in a certain order. This ensures that we obtain the correct final result.
Only after handling operations inside the parentheses do we proceed to addition or subtraction outside the brackets. Following this ordered approach is vital in arriving at the correct solution without errors.

Negative Numbers

Negative numbers can sometimes be tricky, especially when they are wrapped inside parentheses or combined with other negative or positive numbers.
When simplifying expressions, it's important to pay attention to the signs of the numbers involved. For example, in \((4-15)\), it simplifies to \(-11\) because subtracting a larger number from a smaller number gives a negative result.
Another point to note is the subtraction of negative numbers, like \(-(-6)\), which translates to a positive 6. This reversal occurs because subtracting a negative is the same as adding its positive counterpart. Always remember, two negative signs next to each other become positive.
Being comfortable with these rules will help you navigate and solve expressions smoothly.

Parentheses in Algebra

Parentheses in algebra are used to group terms and indicate which operations should be carried out first. They play an essential role in guiding the order of operations.
In the expression \(-5+[(4-15)-(-6)-8]\), we see two levels: brackets and parentheses. The reason to begin with the innermost parentheses is to correctly evaluate the components of the expression from the deepest level of segregation first. This is part of the ordered thinking promoted by the PEMDAS rule.
Once the operations within these groups are simplified, we can then consider the results as single entities while simplifying further up in the expression. It helps to think of solving each part within parentheses or brackets as tackling a mini problem before moving on to the next, ensuring each computation follows the correct sequence.