Problem 9 - Algebra and Trigonometry

Question

Evaluate the arithmetic expression. \(\frac{5+7}{3}-6[12-(17-2 \cdot 3)]\)

Step-by-Step Solution

Answer

The expression evaluates to \(-2\).

Step 1: Simplify the innermost expression

First, simplify the innermost part inside the parentheses: \(17 - 2 \cdot 3\). Calculate the multiplication: \(2 \cdot 3 = 6\). Substitute back: \(17 - 6 = 11\).

Step 2: Simplify the expression inside brackets

Now substitute \(11\) back into the expression: \(12 - 11\). Perform the subtraction: \(12 - 11 = 1\).

Step 3: Apply the brackets with multiplication

Now apply the multiplication outside the brackets: \(-6 \cdot 1 = -6\).

Step 4: Simplify the expression within the fraction

Add the numbers in the numerator: \(5 + 7 = 12\).

Step 5: Simplify the fraction

Divide the result from Step 4 by 3: \(\frac{12}{3} = 4\).

Step 6: Final combination of all simplified parts

Combine the results: \(4 - 6 = -2\).

Step 7: Conclusion

The final result of the expression is \(-2\).

Key Concepts

Order of Operations

Understanding the order of operations is vital when evaluating arithmetic expressions. It's like following a recipe in a precise order to get the perfect cake. In mathematics, we use a standard hierarchy to decide the sequence of operations.

Think of this order in terms of the acronym PEMDAS:

• Parentheses (and brackets) first
• Exponents (or roots)
• Multiplication and Division (left to right)
• Addition and Subtraction (left to right)

Following this order helps to avoid mistakes when simplifying mathematical expressions.
In the original exercise, we start by simplifying within both parentheses and brackets first before moving on to operations like multiplication, division, addition, or subtraction. This guarantees that each step builds correctly on the previous one.

Simplification

Simplification is the process of reducing a mathematical expression to its simplest form. It's like cleaning up a cluttered room to make it more organized and presentable. We simplify step by step, tackling operations in the correct sequence.

In this problem, we first worked on the innermost parentheses to find the simplest form of that specific section. This involved calculating the product within:

• We began with multiplying: \(2 \cdot 3 = 6\),
• then subtracting from 17: \(17 - 6 = 11\).

Simplifying within the brackets came next, leading us to a cleaner form:

• \(12 - 11 = 1\).

Finally, we dealt with the other operations outside, including division and subtraction. At each point, we simplify the expression, making it straightforward and easier to handle. This method allows for accuracy and ensures the final evaluation aligns with mathematical principles.

Mathematical Expressions

Mathematical expressions are like special sentences that use numbers and symbols instead of words. They're a way of communicating mathematical ideas comprehensively. An expression can include operations like addition, subtraction, multiplication, and division, as well as numbers and variables.

When approaching mathematical expressions, it's crucial to understand both what's being asked and how different elements interact. Our original exercise involved various components:

• A fraction: \(\frac{5+7}{3}\),
• A subtraction with a set multiplication within brackets: \(-6[12 - (17 - 2 \cdot 3)]\).

By breaking down expressions into manageable parts, we can handle complex calculations step-by-step, applying our knowledge of order of operations and simplification.
Mathematical expressions form the foundation of math problems in fields ranging from basic arithmetic to advanced calculus, making understanding them essential for solving math-related challenges efficiently.