Problem 35 - Algebra A Combined Function

Question

Simplify each expression. $$ 2[5+2(8-3)] $$

Step-by-Step Solution

Answer

The simplified expression is 30.

Step 1: Solve within brackets

The expression inside the outer brackets is \(5 + 2(8-3)\). Start by solving the expression inside the innermost parentheses: \(8 - 3\).

Step 2: Simplify parentheses

Subtract \(3\) from \(8\) to get \(5\). So the expression now reads \(5 + 2 imes 5\).

Step 3: Multiply

According to the order of operations, perform the multiplication next: \(2 \times 5 = 10\). Now the expression is \(5 + 10\).

Step 4: Add

Finally, add \(5\) and \(10\) to get \(15\).

Step 5: Multiply outside the brackets

Now multiply the entire result by the \(2\) outside the brackets: \(2 \times 15 = 30\).

Key Concepts

Order of Operations

When faced with an expression that requires simplification, it's essential to follow the order of operations. The order of operations is a set of rules that determines the sequence in which operations should be performed to ensure accurate results. It's often remembered by the acronym PEMDAS:

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

Because multiplication and division, as well as addition and subtraction, have the same precedence level, they are to be performed from left to right in the expression.
This order ensures that every mathematician approaches a problem the same way, reducing the chance for errors or misunderstandings. For the expression we have, following PEMDAS means first solving inside the parentheses and then continuing outward, managing multiplication and addition in the sequence required by the operation order rules.

Brackets and Parentheses

Brackets and parentheses are critical in mathematical expressions because they dictate which operations need to be performed first. In the expression provided, we initially encounter both brackets (square) and parentheses (round).
Parentheses are dealt with before any other operations. In this case, \(8 - 3\) within the parentheses needed addressing first. These groupings help clarify the order when multiple operations are present. After solving the innermost parentheses, you then work through the brackets.

• Step inside the innermost parentheses, solving \(8 - 3 = 5\).
• After addressing these, deal with the expression within the square brackets, treating them like regular parentheses.

In this manner, brackets and parentheses help to prioritize operations correctly in complex equations, ensuring precise calculations.

Multiplication and Addition

Once the expressions inside the brackets and parentheses were solved, the next step is to handle multiplication before addition, as dictated by the order of operations.
In our example, after simplifying the expression to \(5 + 2 \times 5\), multiplication comes first. Calculating \(2 \times 5\) yields \(10\). Once multiplication is completed, the expression reduces to simple addition: \(5 + 10\).
These steps show how multiplication has precedence over addition in calculations to maintain accuracy. After multiplication, you can comfortably proceed to the addition step to conclude the simplification of the expression into its simplest form.