Evaluate each expression for $p=-4$, $q=8$, and $r=-10$. $\frac{-(p + 2)^2 - 3r}{2 - q}$

Verified step by step guidance

1. First, substitute the given values into the expression: replace \( p \) with \( -4 \), \( q \) with \( 8 \), and \( r \) with \( -10 \) in the expression \( - (p+2)^2 - \frac{3r}{2} - q \).
2. Next, simplify inside the parentheses: calculate \( p + 2 \) by adding \( -4 + 2 \).
3. Then, square the result from the previous step to get \( (p+2)^2 \).
4. After that, multiply \( 3 \) by \( r \) and then divide by \( 2 \) to simplify the fraction \( \frac{3r}{2} \).
5. Finally, combine all parts of the expression by applying the negative sign to the squared term, subtracting the fraction, and then subtracting \( q \) to simplify the entire expression.

Key Concepts

Order of Operations

The order of operations dictates the sequence in which mathematical operations are performed to ensure consistent results. It follows the PEMDAS rule: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Applying this correctly is essential when evaluating expressions with multiple operations.

Substitution of Variables

Substitution involves replacing variables in an expression with given numerical values. This step is crucial to evaluate expressions numerically. Careful substitution ensures that each variable is correctly replaced before performing arithmetic operations.

Exponents and Squaring

Exponents indicate repeated multiplication of a base number. Squaring a number means raising it to the power of 2. Understanding how to correctly apply exponents, especially when combined with parentheses, is important to accurately simplify expressions.