Evaluate 2(45÷((11-8)^2))-3

Solution:

Step 1: Break down the expression into simpler parts.

Step 1.1: Focus on the denominator first.

Step 1.1.1: Calculate the difference between $11$ and $8$.

$2\times \frac{45}{(11-8{)}^2}-3$

Step 1.1.2: Square the result of $11-8$, which is $3$.

$2\times \frac{45}{3^2}-3$

Step 1.2: Now, divide $45$ by the square of $3$, which is $9$.

$2\times \frac{45}{9}-3$

Step 1.3: Multiply $2$ by the result of the division.

$2\times 5-3$

Step 2: Finally, subtract $3$ from the product of $2$ and $5$.

$10-3$

The final answer is:

$7$

Knowledge Notes:

To solve the given mathematical expression, we need to apply the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This rule dictates the sequence in which the operations should be performed to accurately evaluate the expression.

1. Parentheses: Solve expressions inside parentheses or brackets first. In this case, we calculate $(11-8)$.

2. Exponents: Next, we handle any exponents, which means squaring the result of $11-8$.

3. Multiplication and Division: These operations are performed from left to right. We divide $45$ by the squared result and then multiply by $2$.

4. Addition and Subtraction: Lastly, we perform addition and subtraction from left to right. We subtract $3$ from the result of the multiplication.

In LaTeX, we use the $÷$symbol for division, $\text{Missing superscript or subscript argument}$for exponents, and $\times$for multiplication. It's important to format expressions clearly to avoid confusion, especially when using parentheses to denote division or multiplication operations.