Problem 67 - Algebra for College Students

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: Solve Inside the First Bracket

Begin by simplifying the expression inside the first bracket: \(3 + 2(4 \cdot 1 - 2)\). Start by performing the multiplication: \(4 \cdot 1 = 4\). This gives us: \(3 + 2(4 - 2)\).

Step 2: Simplify the inside of the Parentheses

Next, simplify the subtraction inside the parenthesis: \(4 - 2 = 2\). So our expression becomes \(3 + 2 \cdot 2\).

Step 3: Solve the Multiplication

Now, multiply \(2 \cdot 2\) to get: \(4\). Substitute this into the expression: \(3 + 4\).

Step 4: Finalize the First Bracket

Add the numbers: \(3 + 4 = 7\). So, the simplified first bracket is \(7\).

Step 5: Solve Inside the Second Bracket

Now, simplify the expression inside the second bracket: \(18 - (2 \cdot 4 - 7 \cdot 1)\). Start with the multiplications: \(2 \cdot 4 = 8\) and \(7 \cdot 1 = 7\). The expression becomes \(18 - (8 - 7)\).

Step 6: Solve the Inside Subtraction

Perform the subtraction inside the parenthesis: \(8 - 7 = 1\). The expression now is \(18 - 1\).

Step 7: Finalize the Second Bracket

Subtract \(18 - 1\) to get: \(17\). Thus, the simplified second bracket is \(17\).

Step 8: Multiplication of Simplified Brackets

With both brackets simplified, multiply the two results: \(7 \cdot 17\).

Step 9: Final Calculation

Calculate \(7 \cdot 17\): \(7 \cdot 17 = 119\).

Key Concepts

Order of Operations

In mathematics, the order of operations is a set of rules that determines the sequence in which parts of an expression are calculated. This helps clear up any ambiguity. A common mnemonic to remember the order is PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
In the expression \([3+2(4 \cdot 1-2)][18-(2 \cdot 4-7 \cdot 1)]\), we first focus on operations within the parentheses or brackets, as they take precedence.

• Start with operations inside the innermost parentheses or brackets.
• Perform any operations involving exponents next.
• Then, do multiplication and division from left to right.
• Finally, carry out addition and subtraction from left to right.

By following this order, we ensure that expressions simplify correctly and consistently.

Multiplication

Multiplication is a foundational arithmetic operation that essentially involves adding a number to itself a specified number of times. For instance, when multiplying 4 by 1, you would get 4 because you are effectively adding 4 to itself once.
In our exercise, at the start, we performed:

• \(4 \cdot 1 = 4\)

Later, inside the second bracket, another multiplication appears:

• \(2 \cdot 4 = 8\)
• \(7 \cdot 1 = 7\)

Multiplication precedes addition and subtraction, so it must be completed first, following the order of operations. This step is crucial because it directly affects the result of the equations by effectively compressing terms, which can simplify the overall expression.

Addition

Addition is the process of calculating the total of two or more numbers or amounts. It is one of the fundamental operations in arithmetic that builds the basis for complex mathematics. In the given mathematical expression, addition is used to simplify portions of the expression after multiplication has been performed.
After multiplying in the first bracket, we encountered:

• Result: \(3 + 4 = 7\)

Here, combining 3 and 4 gives us 7, completing the first bracket.
Adding numbers is typically the final step when simplifying expressions after parentheses, multiplication, and any subtractions are resolved, conferring the result for a given bracket or the whole expression.

Subtraction

Subtraction is an arithmetic operation that involves taking one number away from another. In expressions with multiple steps and operations, subtraction usually follows multiplication and division in the order of operations.
In our task, subtraction occurred in several steps:

• Subtraction inside the first bracket: \(4 - 2 = 2\)
• Subtraction inside the second bracket: \(8 - 7 = 1\)

Finally, outside the bracket, the subtraction of the resolved terms continues:

• \(18 - 1 = 17\)

Subtracting often finalizes calculations after multiplications in related parts of an expression, paving the way to reach the simplified form or solution.