Problem 17 - College Algebra

Question

Simplify the given expression. $$ 14 \cdot 3 \div 7-6 $$

Step-by-Step Solution

Answer

The simplified expression equals 0.

Step 1: Multiply

First, multiply the numbers 14 and 3. \[ 14 \cdot 3 = 42 \]

Step 2: Divide

Next, divide the result from the multiplication by 7. \[ \frac{42}{7} = 6 \]

Step 3: Subtract

Finally, subtract 6 from the result of the division. \[ 6 - 6 = 0 \]

Key Concepts

Multiplication and Division

In arithmetic, multiplication and division are fundamental operations used to solve equations. These operations are closely related, similar to how addition and subtraction are. When simplifying expressions, it's important to follow the prescribed order of operations. This usually means addressing multiplication and division before moving on to addition and subtraction. For instance, in the expression \( 14 \cdot 3 \div 7 \), we start by multiplying 14 and 3. The result is 42. Using multiplication first ensures that our calculations adhere to the order of operations, resulting in the correct answer.

• Multiplication essentially means repeated addition. For example, \( 14 \cdot 3 \) can be seen as adding 14 three times.
• Following multiplication, division helps distribute a total into equal parts, making it a form of repeated subtraction.

By dividing 42 by 7, we obtain 6, which readies us for the next step, subtraction. Always keep multiplication and division sequences in order to guarantee accuracy.

Arithmetic Simplification

Simplification in arithmetic involves breaking down complex expressions into simpler forms. This is achieved by dealing with operations step by step as dictated by the order of operations. Each operation contributes to reducing the expression to its most straightforward form. In the case of the expression \( 14 \cdot 3 \div 7 - 6 \), starting with multiplication and division simplifies the numbers involved, leading to a simple subtraction problem.

• Simplification helps to avoid mistakes by narrowing down complex calculations to manageable numbers.
• By working on one operation at a time, we reduce the risk of making errors and ensure clarity in arriving at the correct answer.

After multiplication and division, we're left with the final subtraction. Ensuring each operation is completed correctly is key to achieving a simplified and accurate outcome.

Subtraction in Arithmetic

Subtraction in arithmetic is the method used to find the difference between numbers. This operation is often the final step in simplifying expressions as it builds on the results from prior operations. In the example \( 6 - 6 = 0 \), subtraction follows multiplication and division.

• It is essential to apply subtraction carefully, especially as it is reliant on earlier calculated results in multi-step problems.
• For subtraction, ensure that the numbers used are accurate, as any mistake in previous steps can propagate to this final result.

As seen here, simplifying expressions step-by-step leads us to subtraction, where it's straightforward enough to plainly calculate the difference, resulting in 0. This highlights the importance of correct ordering and solving the arithmetic operations sequentially.