Problem 65 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the order of operations to simplify each expression. $$24 \div \frac{3^{2}}{8-5}-(-6)$$

Step-by-Step Solution

Answer

14

Step 1: Evaluate expression inside the brackets

In the given expression \(24 \div \frac{3^{2}}{8-5}-(-6)\), evaluate the expression inside the brackets first according to the rule of BIDMAS which stands for Brackets, Indices, Division, Multiplication, Addition and Subtraction. So, here in the brackets we get \(8 - 5 = 3\). So the expression now becomes \(24 \div \frac{9}{3} - (-6)\).

Step 2: Evaluate division inside the brackets after previous bracket operation

Now, you consider the division operation inside the fraction. The fraction in the brackets is \(\frac{9}{3}\). On simplifying this fraction, we get \(3\). The expression then becomes \(24 \div 3 - (-6)\).

Step 3: Solve Division

The division operation left is \(24 \div 3 = 8\). So, the expression now becomes \(8 - (-6)\).

Step 4: Solve the remaining operations

Now, resolve the expression \(8 - (-6)\) by subtracting the negative value instead which converts the subtraction operation to an addition operation and yields \(8 + 6 = 14\).

Step 5: Final Answer

The entire expression simplifies to the final result 14.

Key Concepts

BIDMAS

Understanding BIDMAS is crucial for solving math problems correctly. Sometimes called PEMDAS in the US, it outlines the order of operations you should follow when simplifying expressions. BIDMAS stands for:

• Brackets
• Indices (also known as powers or exponents)
• Division
• Multiplication
• Addition
• Subtraction

When you encounter a math problem, always start by solving expressions within brackets. Then, handle indices, which are powers or exponents like square or cube. Following that, execute division and multiplication from left to right. Finally, do addition and subtraction, again from left to right. By adhering to this order, you ensure that every expression is simplified correctly, just as in our example problem where the operation within the brackets was addressed first.

Simplification

Simplification in mathematics is the process of reducing an expression or equation to its simplest form. The goal is to make the expression easier to understand or solve, without changing its value.
Simplify step-by-step by focusing on one operation after another while maintaining the hierarchy outlined by BIDMAS.
In our example, simplification started by evaluating the expression inside the brackets with the operation 8 - 5. After turning the original expression into a simpler form by addressing each step in order, we ultimately found the solution. The key is not to rush, and to break down the problem into manageable parts. This ensures accuracy while making the process less overwhelming.

Division

Division is one of the four basic arithmetic operations and is represented by the symbol \( \div \). It involves splitting a number into equal parts.
When simplifying expressions, always look for division operations and solve them next after any brackets and indices have been addressed.
In our example, after simplifying the fraction within the brackets \( \frac{9}{3} \), we followed BIDMAS and tackled the remaining division operation in the expression: \( 24 \div 3 \). This step successfully reduced the expression, setting the stage for further simplification. Remember, division should always be handled carefully to maintain accuracy.

Negative Numbers

Working with negative numbers can sometimes be tricky, but understanding them is crucial in mathematics. When dealing with negative signs, pay special attention to operations involving subtraction and addition.
A common mistake is misinterpreting the subtraction of negative numbers. Remember that subtracting a negative number is the same as adding its positive counterpart. In the final step of our example \( 8 - (-6) \), the expression converts to \( 8 + 6 \) due to the double negative rule. This ultimately results in 14, clearly showing how handling negative numbers correctly completes the simplification process. Practice handling negative numbers with care, and soon you'll be managing them with confidence.