Problem 67 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ [3+2(4 \cdot 1-2)][18-(2 \cdot 4-7 \cdot 1)] $$

Step-by-Step Solution

Answer

The simplified expression is 119.

Step 1: Simplify Inside the First Parentheses

Start by simplifying the expression inside the first set of parentheses: \(3+2(4 \cdot 1-2)\). Calculate the multiplication first: \(4 \cdot 1 = 4\).This gives us \(3+2(4-2)\). Next, simplify \(4 - 2 = 2\), resulting in \(3+2 \cdot 2\).

Step 2: Evaluate the First Parentheses

Continue to evaluate \(3+2 \cdot 2\). Calculate the multiplication: \(2 \cdot 2 = 4\). Now add \(3 + 4 = 7\). So the expression simplifies to \(7\) in the first set of parentheses.

Step 3: Simplify Inside the Second Parentheses

Now simplify the expression inside the second set of parentheses: \(18-(2 \cdot 4-7 \cdot 1)\).Start by calculating the multiplications: \(2 \cdot 4 = 8\) and \(7 \cdot 1 = 7\). Now simplify \(8 - 7 = 1\), so we have \(18-1\).

Step 4: Evaluate the Second Parentheses

Simplify \(18 - 1\) to get \(17\). So the expression in the second set of parentheses simplifies to \(17\).

Step 5: Combine the Simplified Expressions

Substitute the simplified expressions back into the original expression: \([7][17]\).The expression \([7][17]\) represents the multiplication of the two simplified results: \(7 \cdot 17\).

Step 6: Compute the Final Answer

Finally, multiply the two results: \(7 \cdot 17\). This calculation gives \(119\).So, the simplified expression equals \(119\).

Key Concepts

Order of Operations

In mathematics, solving expressions in the correct sequence is essential. This is known as the order of operations. It ensures that everyone interprets and simplifies expressions uniformly. Usually, this involves a specific sequence:

• First, solve any operations within parentheses or brackets.
• Next, perform any multiplications or divisions from left to right.
• Finally, conduct any addition or subtraction actions from left to right.

Remembering this sequence helps you consistently achieve accurate results. A common way to remember it is by using the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This step-by-step approach provides clarity, especially when faced with complicated mathematical equations.

Parentheses in Expressions

Parentheses play a crucial role in mathematics. They indicate which part of an equation should be solved first. In our given expression, parentheses dictate the primary operations:

• First set: \(3+2(4 \cdot 1-2)\); addressing this tells us what needs priority in this segment.
• Second set: \(18-(2 \cdot 4-7 \cdot 1)\); another operation to tackle while maintaining order.

By solving what's within the parentheses initially, you simplify the equation down to more manageable pieces. This prioritization is fundamental, as it ensures you're handling complex equations methodically. Always remember: parentheses present the first puzzle piece to work out.

Multiplication and Addition

When tackling equations that involve both multiplication and addition, it's crucial to know their priority in operations. Multiplication takes precedence over addition. This rule prevents misinterpretations:

• Upon simplifying \(4 \cdot 1\) to get \(4\), it becomes part of the equation \(3+2(4-2)\).
• Next, calculate \(2 \cdot 2\) to achieve \(4\); this should occur before adding because multiplication comes first.

When done correctly, the simplified form of the equation shows accurate results. Therefore, never rush to add numbers until you confirm all multiplication is complete within your expression.