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

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\(. The expression is \)\frac{3p + 3(4 + p)^3}{r + 8}\(, so after substitution it becomes \)\frac{3(-4) + 3(4 + (-4))^3}{-10 + 8}$.
2. Next, simplify inside the parentheses: calculate \$4 + (-4)\( which simplifies to \)0\(. So the expression inside the cube becomes \)0^3$.
3. Then, calculate the cube: \$0^3 = 0\(. Now the numerator simplifies to \)3(-4) + 3 \times 0$.
4. Simplify the numerator: \$3(-4) = -12\( and \)3 \times 0 = 0\(, so the numerator is \)-12 + 0 = -12$.
5. Simplify the denominator: \(-10 + 8 = -2\). Now the expression is \(\frac{-12}{-2}\). This fraction can be simplified further by dividing numerator and denominator.

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 their given numerical values. This step is crucial for evaluating expressions with variables, allowing you to simplify and calculate the final numerical result accurately.

Exponentiation

Exponentiation is the operation of raising a base number to a power, indicating repeated multiplication. Understanding how to compute powers, especially with parentheses affecting the base, is important for correctly evaluating expressions involving exponents.