Evaluate each expression for p = -4, q = 8, and r = -10. See Example 6. (-(p + 2)² - 3r)/(2 - q)

Verified step by step guidance

1. First, substitute the given values of \( p = -4 \), \( q = 8 \), and \( r = -10 \) into the expression: \( - (p + 2)^2 - 3r \) divided by \( 2 - q \).
2. Rewrite the expression with the substituted values: \( - (-4 + 2)^2 - 3(-10) \) over \( 2 - 8 \).
3. Calculate the value inside the parentheses in the numerator: \( (-4 + 2) = -2 \), then square it to get \( (-2)^2 \).
4. Evaluate the numerator by applying the negative sign outside the square, then subtract \( 3r \) (remember \( r = -10 \)), so compute \( - (-2)^2 - 3(-10) \).
5. Calculate the denominator by subtracting \( q \) from 2: \( 2 - 8 \), then divide the evaluated numerator by this denominator to complete the expression.

Key Concepts

Order of Operations

The order of operations dictates the sequence in which mathematical operations are performed: parentheses first, then exponents, followed by multiplication and division (left to right), and finally addition and subtraction (left to right). This ensures consistent and correct evaluation of expressions.

Substitution of Variables

Substitution involves replacing variables in an expression with their given numerical values. This step is essential to evaluate expressions numerically, allowing the transformation of an algebraic expression into a specific number.

Evaluating Exponents and Negative Signs

When evaluating expressions with exponents, it is important to correctly apply powers to quantities, especially when negative signs and parentheses are involved. For example, squaring a negative number inside parentheses affects the sign of the result, while a negative sign outside the parentheses changes the overall sign after exponentiation.