Problem 68 - Algebra 2

Question

Simplify. $$ 3+(21 \div 7) \times 8 \div 4 $$

Step-by-Step Solution

Answer

The simplified expression equals 9.

Step 1: Perform Division Inside the Parentheses

First, solve the division inside the parentheses: \(21 \div 7\).\[21 \div 7 = 3\]

Step 2: Rewrite the Expression

Now that you have simplified inside the parentheses, rewrite the expression substituting the result.The expression becomes: \[3 + 3 \times 8 \div 4\]

Step 3: Perform the Multiplication

Next, perform the multiplication in the expression: \(3 \times 8\).\[3 \times 8 = 24\]

Step 4: Perform the Division

Now, divide the result of the multiplication by 4: \(24 \div 4\).\[24 \div 4 = 6\]

Step 5: Perform the Addition

Finally, add the result from Step 4 to the initial 3: \(3 + 6\).\[3 + 6 = 9\]

Key Concepts

Simplification of Expressions

In mathematical exercises, especially when dealing with multiple operations, simplification is a key process. It helps in reducing the expression to its most basic form. The goal is to make calculations more straightforward and less prone to errors. Simplification often involves following standard procedures like solving inside parentheses first, as they dictate which operations are prioritized according to the order of operations. In our exercise, the expression starts with a parenthesis which signifies performing the operation inside it first. By handling the division within the parentheses, we reduce complex expressions to simpler numeric values. To effectively simplify an expression:

• First, address any operations within parentheses. This is always the first step in the order of operations.
• Next, execute any exponents, though they are not present in this exercise.
• Proceed with multiplication and division from left to right, followed by addition and subtraction.

Breaking down expressions into simpler parts ensures accuracy and makes the solution process manageable.

Division and Multiplication

Once the parentheses are addressed, division and multiplication come next in the order of operations. They hold equal priority, so they are performed from left to right as they appear in the expression.In this exercise, following the simplification inside the parenthesis, we encounter a multiplication followed by a division:

• First, perform the multiplication: calculate the product of the numbers that follow one another, as seen with the expression changing from
  \(3 \times 8\).
• Next, carry out the division: take the result from the previous multiplication and divide it by the next number, such as with \(24 \div 4\).

Multiplication and division are considered at the same level, meaning the order in which these operations are performed is dictated solely by their position when reading from left to right. Following this rule correctly simplifies computations significantly.

Addition and Subtraction

After addressing multiplication and division, we reach the stage where addition and subtraction are handled. Much like their counterparts, these operations are of equal precedence but are completed after any multiplication or division. In this exercise, after successfully dividing the product from the multiplication, we add the remaining figures:

• Firstly, identify any resulting values from prior computations, like the number 6 which comes from our division operation.
• Next, add this result to the earlier figure in the expression that didn’t partake in previous operations, in this case, the 3.

The final arithmetic step in simplifying the expression is simple addition. The ease comes from having potentially complex expressions reduced step by step earlier, making the final sum factor quite straightforward.