Question

Explain how to evaluate an expression. $$2 r^{2}+4 s-4 ; r=-3, s=5$$

Short answer

Answer: 34

Step-by-step solution

Substitute the values of r and s

Replace the variables r and s in the expression with their given values, which are r = -3 and s = 5: $$2(-3)^2 + 4(5) - 4$$.

Simplify the expression according to the order of operations

First, we need to calculate the value inside the parentheses and then exponentiation, followed by multiplication and addition/subtraction. 1. Start by calculating the value of the exponent: $$(-3)^2 = 9$$. 2. Now substitute this value back into the expression: $$2(9) + 4(5) - 4$$. 3. Perform multiplication: $$18 + 20 - 4$$. 4. Perform addition and subtraction from left to right: $$18 + 20 - 4 = 34$$. The evaluated expression is 34.

Key concepts

Order of Operations

Understanding the order of operations is crucial when it comes to evaluating mathematical expressions. It's the set of rules that dictates which calculations to perform first to obtain the correct result. A common acronym to remember the order is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

When evaluating an expression, always start by simplifying the operations inside any parentheses. Next, resolve any exponents or powers. Then, tackle multiplication and division as they appear, working from left to right. Finally, perform addition and subtraction, also from left to right. For example, given the expression \(2 r^{2}+4 s-4\), and when substituting variables with values, use PEMDAS to avoid mistakes. Failing to follow this order can lead to incorrect answers and confusion in more complex algebraic expressions.

Substitution in Algebra

Substitution is a fundamental technique in algebra used to evaluate expressions with one or more variables. This method involves replacing variables with their corresponding values to simplify an expression into a solvable numeric equation. When substituting, it's important to replace each instance of the variable with the given value.

For instance, in the expression \(2 r^{2}+4 s-4\), we have the variables \(r\) and \(s\) with given values \(r=-3\) and \(s=5\). By substituting these values into the original expression, we obtain a numerical expression that can then be simplified using the order of operations. It's essential to write the substituted values in parentheses to clearly delineate them, particularly when they are negative numbers or complex expressions themselves. This practice ensures accuracy during the simplifying process.

Exponentiation

Exponentiation is an arithmetic operation involving a base and an exponent. The base is the number being multiplied by itself, and the exponent tells us how many times to multiply the base. For example, \(3^2\) means to multiply 3 by itself twice, resulting in 9.

In evaluating the expression \(2 r^{2}+4 s-4\) when \(r=-3\), the term \(r^{2}\) tells us to square the value of \(r\). Thus, \( (-3)^2 = -3 \times -3 = 9\). When dealing with negative bases, it's important to remember that an even exponent will result in a positive outcome, while an odd exponent will keep the negative sign. After exponentiation, you can proceed with the expression simplification through multiplication, addition, or subtraction as per the order of operations.