Problem 37 - Elementary Algebra

Question

Simplify. $$ 5 \cdot 215-(12) 3 $$

Step-by-Step Solution

Answer

The simplified result is 1039.

Step 1: Identify the Expression

The given expression to simplify is: \[5 \cdot 215 - (12) 3\]

Step 2: Simplify the Multiplication Inside the Parentheses

Calculate the product inside the parentheses, which is \( (12) 3 \). \[12 \times 3 = 36\]

Step 3: Substitute and Restate the Expression

Replace \((12) 3\) with \(36\) in the original expression, resulting in:\[5 \cdot 215 - 36\]

Step 4: Simplify the Remaining Multiplication

Calculate the product \( 5 \cdot 215 \):\[5 \times 215 = 1075\]

Step 5: Perform the Subtraction

Subtract \(36\) from \(1075\):\[1075 - 36 = 1039\]

Step 6: State the Simplified Result

The expression simplifies to:\[1039\]

Key Concepts

Multiplication in Algebra

Multiplication in algebra is a fundamental operation that simplifies expressions by combining numbers or variables. When we multiply, we are essentially adding a number to itself repeatedly. In the exercise provided, multiplication is used twice.

• The first multiplication, \( 12 \times 3 \), simplifies the expression inside the parentheses. This helps to simplify steps by reducing complex expressions into simpler numerical values.
• The second multiplication, \( 5 \times 215 \), combines numbers outside the original parentheses. This step further simplifies the expression by producing larger, yet manageable numbers.

These types of simplifications make it easier to evaluate and understand algebraic expressions. By breaking down the expression into straightforward calculations, we can find solutions quickly and efficiently.

Order of Operations

The order of operations is critical in mathematics to ensure consistent results when simplifying expressions. The acronym PEMDAS can help you remember this order: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).For the problem at hand, the order of operations guides us to:

• First, address operations inside the parentheses. We simplified \((12 \times 3)\) first as given in the problem.
• Then, proceed with multiplication across the remaining expression. We evaluated \(5 \times 215\) next.
• Finally, we perform subtraction on these resulting numbers to further simplify the expression to its final form.

By following PEMDAS, you'll ensure each step is done in the correct order, avoiding potential errors and achieving the correct results swiftly.

Subtraction in Algebra

Subtraction in algebra involves removing a specific value from another to complete the simplification process after multiplication. Subtraction helps to simplify expressions and bring them down to single numerical values. For example, in the given problem:

• Once multiplication is complete, the resulting numbers are then reduced by subtraction: \(1075 - 36\).
• This subtraction merges all previous operations into one final simplified number, \(1039\) in this case.

Being methodical in how you approach subtraction ensures that results are precise and the final outcome is correct. Practice recognizing when subtraction comes into play, especially after completing multiplication or division parts of expressions.