Problem 8 - Elementary Algebra

Question

Use the order of operations to find each value. $$4+3[2+3(1+8 \div 4)]$$

Step-by-Step Solution

Answer

Answer: The value of the expression is 37.

Step 1: Perform inside parentheses operations

We perform the innermost operations first, which are enclosed in parentheses. In this case, that's the division: $$4+3[2+3(1+8 \div 4)] \Rightarrow 4+3[2+3(1+2)]$$

Step 2: Continue with brackets operations

Now, we'll perform the operation inside the brackets, starting with the operation within the parentheses: $$4+3[2+3(1+2)] \Rightarrow 4+3[2+3(3)]$$

Step 3: Perform multiplication inside brackets

There is still a multiplication operation inside the brackets: $$4+3[2+3(3)] \Rightarrow 4+3[2+9]$$

Step 4: Complete brackets operations

Now perform the remaining addition operation inside the brackets: $$4+3[2+9] \Rightarrow 4+3[11]$$

Step 5: Perform final multiplication

Finally, perform the remaining multiplication operation: $$4+3[11] \Rightarrow 4+33$$

Step 6: Perform final addition

Last, perform the remaining addition operation: $$4+33 = 37$$ So, the value of the given expression is 37.

Key Concepts

Parentheses

Parentheses are the first component in the order of operations, also known as PEMDAS/BODMAS. They function as guides indicating which calculations should be carried out first. For example, in the expression \(1+8 \div 4\), the operation within parentheses should be solved before handling any operations outside of them.
In our exercise, the innermost part of the expression was \(1+8 \div 4\). According to the order of operations, we solve \(8 \div 4\) first, resulting in \(1+2\).

• The division inside the parentheses was prioritized.
• Understanding parentheses can significantly simplify complex expressions.
• The use of parentheses helps organize calculations effectively.

Remember, always tackle what’s inside parentheses before moving on to other operations. This ensures nothing gets overlooked as calculations become more complex.

Brackets

Brackets follow parentheses in the hierarchy of operations. They enclose expressions and often contain multiple operations. In our expression, after handling the content inside the parentheses, the focus shifted to the brackets: \(4+3[2+3(1+2)]\).
Within the brackets, there is another layer of complexity. Here, you must resolve everything inside before moving outward. Applying the order of operations again, we first handle operations inside the parentheses, then move to addition: \(2+3(3)\).

• Brackets allow for additional layers of grouping in mathematical expressions.
• They clarify the sequence of operations.
• Always work from the innermost part outward.

This method provides clarity to solve each piece effectively without losing track of the bigger calculation plan.

Multiplication

Multiplication steps often come after parentheses and within brackets, unless specified otherwise by the expression's structure. When performing multiplication, it follows that the multiplication inside any grouping symbol is prioritized.
In the expression \(4+3[2+3(3)]\), multiplication within the brackets was next. This involved calculating \(3 \times 3\) to get \(9\).

• Remember that multiplication is performed before addition as per the order of operations.
• Multiplication within brackets or parentheses should be completed before those outside.
• It allows for combining numbers and forms new terms.

Understanding order-related multiplication enables tackling expressions systematically and reduces errors in sequential operations.

Addition

Addition is typically performed last unless otherwise indicated by parentheses or brackets. Completing addition outside of inner groupings finishes the sequence of operations.
After completing all operations involving multiplication and division, as in our given expression \(4+3[11]\), the final arithmetic step is carrying out the simple addition of 4 and 33, producing the final result, 37.

• To ensure accuracy, save addition for last unless enclosed by brackets or parentheses.
• Addition compiles all the intermediary results from prior operations.
• Make sure to follow through the order of additions correctly.

This is why in most complex problems, the addition outside of parentheses and brackets serves as the final step, declaring the culmination and solution of the entire expression.