Problem 68 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ 3[4(6+7)]+2[3(4-2)] $$

Step-by-Step Solution

Answer

The simplified expression is 168.

Step 1: Solve Inside the Parentheses

Start with simplifying the expressions inside the parentheses. First simplify the expression inside the first bracket: \(6 + 7\), which equals \(13\). Next, simplify the expression inside the second bracket: \(4 - 2\), which equals \(2\). This gives us the expression: \[ 3[4(13)]+2[3(2)] \]

Step 2: Solve Inside the Brackets

Next, perform the multiplications inside each set of brackets. For the first set of brackets: \(4 \times 13 = 52\). For the second set of brackets: \(3 \times 2 = 6\). This simplifies our expression to: \[ 3[52] + 2[6] \]

Step 3: Perform the Remaining Operations

Now, perform the remaining multiplications. Multiply the first bracket: \(3 \times 52 = 156\). Multiply the second bracket: \(2 \times 6 = 12\). This updates our expression to: \[ 156 + 12 \]

Step 4: Simplify the Final Expression

Finally, add the results from the previous step: \(156 + 12 = 168\). Thus, the simplified expression is 168.

Key Concepts

Order of Operations

One essential concept for simplifying numerical expressions is the "order of operations," commonly abbreviated as PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This sequence dictates the order in which each operation should be performed to ensure consistent and accurate results. Without following this order, numerical expressions could yield different outcomes depending on how they are approached.

For example, in the expression \(3[4(6+7)]+2[3(4-2)]\), the operations must be performed in a specific sequence:

• First, handle operations within parentheses.
• Then, proceed with any multiplication or division.
• Finally, conduct any addition or subtraction.

By strictly following the order of operations, we ensure that each step builds correctly upon the last, preventing errors and confusion.

Parentheses in Math

Parentheses are used in mathematical expressions to encapsulate certain parts of the equation that need to be prioritized. They signal to the mathematician to first resolve the operations inside them before dealing with anything outside.

Consider the expression presented in the exercise: \(3[4(6+7)]+2[3(4-2)]\). Here, parentheses surround \(6+7\) and \(4-2\). According to the order of operations, these calculations must be completed first. In this case:

• \(6+7\) simplifies to \(13\)
• \(4-2\) simplifies to \(2\)

Once these inner expressions are simplified, the parentheses serve as a guide to simplify the larger expression outside the parenthetical group. This helps maintain clarity and prevent mistakes when solving complex problems.

Arithmetic Simplification

Arithmetic simplification entails reducing a numerical expression to its simplest form. This process often involves operations such as addition, subtraction, multiplication, and sometimes division. Often, simplification continues and combines numbers to obtain the simplest result possible.

In the exercise problem: \(3[4(13)]+2[3(2)]\), once we resolve the expressions inside the parentheses and brackets, the next step is to simplify through multiplication and addition:

• First, compute the multiplications: \(4 \times 13 = 52\) and \(3 \times 2 = 6\).
• This leads us to simplify further: \(3[52] + 2[6]\) leads to \(156 + 12\).
• Lastly, concluding with an addition yields \(168\).

Each step simplifies the expression into something more manageable, ultimately resulting in the clearest, most concise form possible.