Problem 58 - Elementary Algebra

Question

Use the order of operations to simplify the quantities for the following problems. $$ (4+3)^{2}+1 \div(2 \cdot 5) $$

Step-by-Step Solution

Answer

Answer: The simplified value of the expression is 49.1.

Step 1: Parentheses/Brackets

According to the order of operations, we need to first deal with the parentheses. We have: $$(4+3)^2 +1 \div(2\cdot 5)$$ In the parentheses, we have 4+3, which is 7.

Step 2: Exponents

Next, we need to deal with the exponent. $$7^2 + 1 \div(2 \cdot 5)$$ Here, 7 squared is equal to 49.

Step 3: Multiplication and Division

Now, we move on to multiplication and division, working from left to right. $$49 + 1 \div 10$$ Here, we have 1 divided by 10, which is 0.1.

Step 4: Addition and Subtraction

Finally, we move on to addition, the last operation. $$49 + 0.1$$ Adding these values gives us the final simplified result of: $$49.1$$

Key Concepts

Parentheses

When simplifying expressions using the order of operations, the first task is to tackle any parentheses in the expression. Parentheses group numbers and operations to dictate which parts of the expression should be handled first. In our original exercise,
we see \[(4+3)^2 + 1 \div (2 \cdot 5)\] in which the expression inside the parentheses is \((4 + 3)\). Solving this gives us \(7\), shifting the entire expression to \[7^2 + 1 \div (2 \cdot 5)\]. This step ensures that any operations inside the parentheses are completed before moving on to other operations.

• Important note: Always complete operations inside nested parentheses inner-to-outer if present.

Exponents

The next step in the order of operations after dealing with parentheses is to handle any exponents. Exponents represent repeated multiplication of a number by itself.
For example, \(7^2\) means \(7\times7\), which simplifies to \(49\). Using our expression from the previous step, replace the exponent with its calculated value.
The expression now changes from \[7^2 + 1 \div (2 \cdot 5)\] to \[49 + 1 \div (2 \cdot 5)\]. Handling exponents ensures their calculations precede multiplication, division, addition, and subtraction, aligning with the proper hierarchy.

• Remember: Exponents must be resolved before moving on to other operations.

Multiplication and Division

After addressing parentheses and exponents, we proceed with multiplication and division. These operations have equal priority,
so we process them from left to right as they appear in the expression. In our problem, \[49 + 1 \div (2 \cdot 5)\],focus first on the division \(1 \div (2 \cdot 5)\). Calculate inside the parentheses as \((2 \cdot 5) = 10\), then \(1 \div 10\) results in \(0.1\). Complete this before other operations similar in precedence.
This adjustment results in \[49 + 0.1\], setting the expression up for the final steps.

• Important: Always work multiplication and division left to right, the order they appear in the expression, ensuring proper calculation each time.

Addition and Subtraction

The written expression simplifies to its final form with addition and subtraction, the last operations performed in the order of operations. These, like multiplication and division, also hold equal precedence and should be evaluated from left to right.
In our example, should now be \(49 + 0.1\),simply add these two numbers together:
The outcome is a clean, straightforward \(49.1\), completing the simplification process.

• Always perform addition and subtraction in order of appearance, ensuring no missteps are made along the way.
• After completing these steps, make sure to double-check your work for potential errors in simplification.