Problem 5 - College Algebra

Question

Simplify the given expression. $$ 6 \div 2-\left(81 \div 3^{2}\right) $$

Step-by-Step Solution

Answer

The simplified expression is -6.

Step 1: Evaluate the Parenthesis

First, we need to simplify the expression within the parentheses: \(81 \div 3^2\). Start by calculating \(3^2\), which is \(9\). Substitute \(9\) for \(3^2\) in the expression inside the parentheses. The new expression inside the parentheses is now \(81 \div 9\).

Step 2: Simplify the Division Inside Parentheses

Next, perform the division inside the parentheses: \(81 \div 9 = 9\). So, \(81 \div 3^2\) simplifies to \(9\).

Step 3: Simplify the Subtraction

Substitute the result from Step 2 back into the entire expression: \(6 \div 2 - 9\). Now simplify \(6 \div 2\).

Step 4: Perform the Remaining Division

Calculate \(6 \div 2 = 3\). So the expression becomes \(3 - 9\).

Step 5: Conclude with Subtraction

Finally, perform the subtraction: \(3 - 9 = -6\). The simplified expression is \(-6\).

Key Concepts

Order of Operations

Understanding the order of operations is crucial when simplifying expressions in mathematics. This order is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This means that operations must be performed in this specific order to achieve the correct result.

In our expression \[6 \div 2 - (81 \div 3^{2})\]we begin by solving the parentheses first, as indicated by PEMDAS. This ensures that the entire expression is approached methodically, and each element is handled in the right sequence. Following this process prevents mistakes that often occur from completing operations in the wrong order.

Division

Division is a mathematical operation where one number, called the dividend, is divided by another, called the divisor, to get a result known as the quotient. In our problem, division appears multiple times. First, inside the parentheses with the expression
\[81 \div 9\]we divide 81 by 9 to get 9. This operation simplifies the expression inside the parentheses to a single number, which can then be replaced in the original equation.

In the final steps of our solution, division occurs again with 6 divided by 2:
\[6 \div 2 = 3\]Performing division accurately helps transform complex expressions into simpler calculations that are easier to manage. Always ensure you complete any divisions in the correct order as specified by the order of operations, particularly when they appear in parentheses.

Exponents

Exponents are used to express repeated multiplication of a number by itself. For example, in the expression \(3^{2}\) within our problem, the number 3 is multiplied by itself:
\[3 \times 3 = 9\]This calculation is necessary before we can proceed with any operations involving division or subtraction. By simplifying the exponent to its value, we substitute it back into the original expression, which makes further calculations clearer and more straightforward.

• Exponents help to simplify numerical expressions.
• They need to be calculated early in the order of operations.

By understanding exponents, you gain the ability to handle more complex problems and express large numbers in a compact form.

Subtraction

Subtraction involves taking one number away from another. In our final step, once we've dealt with the other operations, we perform the subtraction:
\[3 - 9 = -6\]The result is negative because the number being subtracted (9) is greater than the number it's subtracted from (3). Understanding subtraction is essential for interpreting results correctly, particularly when the operation yields negative results.

• Subtraction needs to be done after resolving parentheses and exponents.
• Consider the sign of your answer carefully, especially if subtracting from a smaller number.

Subtraction might seem basic compared to other operations, but it plays a critical role in reaching a problem's final solution, as it did in revealing the final answer of \(-6\) for our expression.