Evaluate each expression for p=-4, q=8, and r=-10. -p² -7q +r²

Verified step by step guidance

1. Identify the given expression and the values of the variables: The expression is \(-p^{2} - 7q - r^{2}\), with \(p = -4\), \(q = 8\), and \(r = -10\).
2. Substitute the given values into the expression: Replace \(p\) with \(-4\), \(q\) with \(8\), and \(r\) with \(-10\) to get \(-(-4)^{2} - 7(8) - (-10)^{2}\).
3. Calculate the squares: Compute \((-4)^{2}\) and \((-10)^{2}\) separately. Remember that squaring a negative number results in a positive number.
4. Multiply and simplify each term: Multiply \(7\) by \(8\), and apply the negative signs to the squared terms as indicated in the expression.
5. Combine all the terms by performing the addition and subtraction to simplify the expression fully.

Key Concepts

Order of Operations

The order of operations dictates the sequence in which mathematical operations are performed, typically remembered by PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures consistent and correct evaluation of expressions, especially when exponents and multiple operations are involved.

Evaluating Expressions with Variables

Evaluating expressions involves substituting given values for variables and simplifying the resulting numerical expression. Careful substitution and simplification are essential to avoid errors, particularly when variables are raised to powers or combined with coefficients.

Exponents and Squaring Negative Numbers

Squaring a number means multiplying it by itself. When squaring negative numbers, the result is always positive because a negative times a negative equals a positive. Understanding this is crucial to correctly evaluate terms like -p², where the placement of the negative sign affects the outcome.