Problem 88 - Elementary Algebra

Question

Experiment with the order of operations on a calculator and share your results.

Step-by-Step Solution

Answer

The result is 13 when correctly applying PEMDAS.

Step 1: Identify the Mathematical Expression

Before using a calculator to experiment with the order of operations, we need a mathematical expression to evaluate. Let's use the expression: \(3 + 5 \times 2\).

Step 2: Review the Order of Operations

In mathematics, the correct order of operations is Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This is often remembered using the acronym PEMDAS.

Step 3: Calculate Without Following PEMDAS on Calculator

Enter the expression into the calculator as is, without considering the order of operations: \(3 + 5 \times 2\). The calculator will process this input as if it’s reading from left to right, resulting in \(3 + 5 = 8\), followed by \(8 \times 2 = 16\).

Step 4: Apply PEMDAS Manually

Apply the order of operations manually. According to PEMDAS, you should first perform the multiplication: \(5 \times 2 = 10\). Then, perform the addition: \(3 + 10 = 13\).

Step 5: Use Calculator Correctly

Now, enter the expression into the calculator correctly considering the order of operations. Use parentheses to ensure the operations are conducted correctly: enter it as \(3 + (5 \times 2)\). Your calculator should now display 13, matching the PEMDAS solution.

Step 6: Analyze the Results

The result from incorrect operation order (16) differs from the correct order using PEMDAS (13). This highlights the importance of using parentheses for clarification in calculations involving multiple operations.

Key Concepts

PEMDAS

PEMDAS is a super handy mnemonic that helps you remember the order to tackle mathematical operations. It's short for Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right), which tells you precisely how to approach any complicated expression.

When working with numbers, always follow PEMDAS to avoid mistakes. Ignore it, and you might mix up your sequence of operations, leading to a wrong answer. Many calculators just read left to right unless instructed otherwise, which can cause incorrect results. To ensure accuracy, practice using PEMDAS on paper first before typing your numbers into a calculator.

• Parentheses: Solve what's inside first.
• Exponents: Simplifying powers or roots next.
• Multiplication and Division: Process from left to right.
• Addition and Subtraction: Complete from left to right.

Parentheses

Parentheses are like a bright spotlight marking what needs to be done first in your math problem. They prioritize certain operations, overriding the standard order. If you're seeing parentheses, it means you need to solve anything inside them before tackling the rest.

Using parentheses can change the outcome dramatically. For instance, with the expression \(3 + 5 \times 2\). If you throw in parentheses around the multiplication like so: \(3 + (5\times 2)\), the answer changes from 16 to 13, as it guides you to handle the multiplication part first.

It's crucial to place them properly:

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• Watch out for nested parentheses (parentheses inside parentheses).

Parentheses bring clarity to complicated calculations, ensuring everyone reads the expression the same way.

Multiplication

Multiplication is a main player in the order of operations. According to PEMDAS, it should be carried out right after parentheses and exponents, but importantly, before addition and subtraction. This is key in maintaining the logical sequence in solving any math problem.

In the example expression \(3 + 5 \times 2\), the multiplication \(5 \times 2 = 10\) comes first, not the addition. Ignoring this rule often leads to incorrect results as shown with 16 being wrongly calculated instead of the right answer, 13, when following correct PEMDAS principles.

Tips for synchronous learning include:

• Flip the sequence: Wrong order produces wrong results.
• Combine multiplication with division first before moving to other operations.

By focusing on multiplication early, you ensure that calculations are precise.

Addition

Addition, being part of the final steps in PEMDAS, comes after you've wrapped up any operations involving parentheses, exponents, multiplication, and division. It's a simple act of bringing numbers together to find their total sum.

Taking our example \(3 + (5 \times 2)\), after calculating the multiplication \((5 \times 2 = 10)\), we add 3, which gives us the final result of 13. If this addition were done before multiplication, it derails the whole operation sequence.

Remember especially:

• Work out all previous operations before jumping into addition.
• Add numbers left to right when dealing with multiple additions.

Ensuring that addition is done at the right time ensures each number is correctly summed up in all expressions.