Let's break down the expression and solve it step by step, applying the proper order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, and Addition/Subtraction).

The expression is:

$13 \div \left( 6.5 - \left| -3 - \frac{1}{4} \right| \right) \cdot \left( (-3)^2 - 9 - \frac{1}{4} \right)$

Step 1: Evaluate inside the absolute value

We first simplify the absolute value expression inside the parentheses: $-3 - \frac{1}{4} = -3.25$ $\left| -3.25 \right| = 3.25$

Now, update the expression: $13 \div \left( 6.5 - 3.25 \right) \cdot \left( (-3)^2 - 9 - \frac{1}{4} \right)$

Step 2: Simplify the terms inside the parentheses

Now, simplify $6.5 - 3.25$: $6.5 - 3.25 = 3.25$

Next, simplify the second part $(-3)^2 - 9 - \frac{1}{4}$: $(-3)^2 = 9$ $9 - 9 - \frac{1}{4} = 0 - \frac{1}{4} = -\frac{1}{4}$

So, the expression becomes: $13 \div 3.25 \cdot -\frac{1}{4}$

Step 3: Perform the division and multiplication

First, calculate $13 \div 3.25$: $13 \div 3.25 = 4$

Then, multiply by $-\frac{1}{4}$: $4 \cdot -\frac{1}{4} = -1$

Final Answer:

The expression evaluates to $-1$.

Let me know if you want more details or if you have any questions!

Related Questions:

1. What is the importance of absolute value in an expression?
2. How do you simplify expressions with exponents and negative numbers?
3. What are common mistakes when applying PEMDAS?
4. How do you convert mixed fractions into decimals or improper fractions?
5. Why does division by fractions work the way it does?

Tip:

Always double-check operations within parentheses first to ensure correct application of the order of operations!