Problem 9 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ 48-3\left[\frac{1+17}{6}\right] $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is 57.

Step 1: Evaluate expression inside parentheses

Evaluate the expression inside the parentheses: (1+17) $$1+17 = 18$$ The expression becomes: $$ 48-3\left[\frac{18}{6}\right] $$

Step 2: Perform the division

Perform the division inside the square brackets: $$\frac{18}{6}=3$$ The expression becomes: $$ 48-3[3] $$

Step 3: Perform the multiplication

Perform the multiplication inside the square brackets: $$-3[3]=-9$$ The expression becomes: $$ 48-(-9) $$

Step 4: Perform the addition

Perform the addition: $$48+9=57$$ The expression is simplified to: $$ 57 $$

Key Concepts

Order of Operations

Order of operations is a fundamental concept in algebra and arithmetic that determines the sequence in which operations should be performed in an expression. The order is often remembered using the acronym PEMDAS, which stands for:

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

In the given exercise, we start by addressing the operation inside the parentheses, which is adding 1 and 17. This is because parentheses have the highest priority in the order of operations. Following this, any division or multiplication takes precedence over addition and subtraction unless they are inside parentheses. This systematic approach ensures that calculations are accurate and consistent. By following PEMDAS, we ensure that the order of computation leads us to the correct simplified result.

Arithmetic Operations

Arithmetic operations are the basic processes of addition, subtraction, multiplication, and division that form the foundation of maths. In our exercise, each of these operations plays a role:

• Addition: The initial step inside the parentheses involves adding 1 and 17.
• Division: This operation comes next according to the order of operations when simplifying the fraction \(\frac{18}{6}\).
• Multiplication: Following division, we multiply 3 by the result obtained from the division within the square brackets.
• Subtraction: Finally, subtraction occurs, although a negative multiplication result effectively turns this into an addition as seen in the final steps.

These operations must be performed in the correct order to simplify the expression properly, transforming it step by step into a single, concise value.

Algebraic Simplification

Algebraic simplification involves the process of reducing expressions to their simplest form without changing their value. This typically includes combining like terms, reducing fractions, and eliminating unnecessary parts of an expression.

In this problem, simplification begins with resolving the innermost operations following the order of operations. Starting with adding and then dividing the result within brackets reduces complexity, step by step, until we arrive at a simple arithmetic operation of 48 minus 9. Simplification helps in solving math problems efficiently and makes it easier to comprehend and communicate mathematical ideas. By breaking down a problem into smaller parts, students can simplify complex expressions with ease, making the entire process much more manageable.