Problem 90 - Elementary Algebra Within Reach

Question

Evaluate the expression. $$34-[54-(-16+4)+6]$$

Step-by-Step Solution

Answer

-38

Step 1: Simplify Inside Parentheses

First, find the result of the operation inside the parentheses: -16 + 4 yields -12.

Step 2: Simplify Inside Square Brackets

Next, find the result of the operation inside the square brackets by subtracting -12 and adding 6 to 54: 54 - (-12) + 6 yields 72.

Step 3: Evaluate the Final Expression

Finally, subtract 72 from 34 to get: 34 - 72 equals -38.

Key Concepts

Parentheses

In arithmetic expressions, parentheses play a crucial role in determining the order of operations. They indicate which parts of an expression should be evaluated first. Without them, expressions become ambiguous and can yield incorrect results.

Here's how you handle parentheses:

• Identify the innermost set of parentheses first and simplify the expression inside completely before moving on.
• Perform arithmetic operations within parentheses according to the standard order of operations: addition and subtraction from left to right.

By following these steps, arithmetic expressions become more manageable and error-free.

Brackets

Brackets, known as square brackets, are used to organize expressions and indicate operations that should be performed next, just after parentheses. They often appear in more complex expressions alongside other types of grouping symbols like curly braces.

Consider the expression given in the task — remember the use of brackets to isolate major sections of the expression. After simplifying expressions inside parentheses, the computation within brackets follows. This ensures a logical, step-by-step simplification of the entire expression. Treating brackets correctly ensures that operations are performed in the right order, maintaining balance in mathematical expressions.

Arithmetic Expressions

Arithmetic expressions are mathematical phrases that involve numbers and operators like "+", "-", "*", and "/". The aim is to find their overall value using systematic steps.

Key points when evaluating an arithmetic expression include:

• Identify and compute operations inside parentheses and brackets first.
• Understand the correct order: parentheses, exponents (if any), multiplication or division, then addition or subtraction. This order is often remembered with the acronym PEMDAS.
• Seamlessly substitute back simplified segments of the expression and continue solving.

Approaching arithmetic expressions methodically prevents errors and leads to correct results.

Negative Numbers

Handling negative numbers requires careful attention. They often change the sign of the overall result when involved in subtraction or addition.

With negative numbers, remember these simple rules:

• Subtracting a negative number is equivalent to adding its positive equivalent. For instance, "Subtract -12" becomes "+12".
• Adding two negative numbers results in a negative sum, while subtracting one negative from another can yield a positive difference.

In the context of the provided exercise, remember: negative signs need careful treatment within and outside parentheses and brackets to ensure they don’t alter the final result incorrectly.