Problem 48 - Algebra for College Students

Question

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. \(2(a+b)^{2}, \quad a=6\) and \(b=-1\)

Step-by-Step Solution

Answer

The evaluated expression is 50.

Step 1: Substitute Values into the Expression

To evaluate the expression \(2(a+b)^2\), first substitute the given values of \(a\) and \(b\) into the expression. We have \(a = 6\) and \(b = -1\). Thus, substituting these values, we get: \[2((6) + (-1))^2\] which simplifies to \[2(6 - 1)^2\].

Step 2: Simplify the Expression Inside the Parentheses

Now, simplify the expression inside the parentheses: \((6 - 1)\). Calculate: \[6 - 1 = 5\]. The expression now is \[2(5)^2\].

Step 3: Calculate the Square of the Expression

Next, calculate the square of the number inside the parentheses: \(5^2\). Therefore: \[5^2 = 25\]. The expression becomes \[2 imes 25\].

Step 4: Perform the Multiplication

Finally, perform the multiplication to evaluate the expression: \[2 imes 25 = 50\]. Thus, the evaluated expression simplifies to 50.

Key Concepts

Substitution Method

The substitution method is a crucial technique for simplifying algebraic expressions. It involves replacing variables with specific given values. In our exercise, we started with the expression \(2(a+b)^2\) and replaced \(a\) with 6 and \(b\) with -1.

• Identify the variables: Here, our variables are \(a\) and \(b\).
• Insert the values: We substitute \(a = 6\) and \(b = -1\) into the expression.

After substitution, the expression becomes \(2((6) + (-1))^2\). This method allows us to transform a general expression into a numerical expression that we can further simplify and solve. Remember, keeping track of positive and negative signs is vital during substitution, as they influence the simplification process.

Evaluating Expressions

Evaluating an expression involves simplifying it down to a single numerical value. Once the substitution is complete, the next step is simplifying the expression.

• First, simplify inside the parentheses: After substitution, we had \(2(6 - 1)^2\). By simplifying inside the parentheses, you calculate \(6 - 1\), resulting in 5.
• Second, simplify `powers` and `roots`: The expression then becomes \(2(5)^2\). Evaluate \(5^2\) to get 25.
• Finish with multiplication: Conclude by multiplying any coefficients. Multiply 2 by 25 to obtain 50. Thus, \(2 \times 25 = 50\).

This ordered process ensures accurate simplification and evaluation of the entire expression.

Order of Operations

Order of operations is a standard rule that dictates the correct sequence to evaluate a mathematical expression. It is commonly abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

• Parentheses: Always start by simplifying expressions inside parentheses. In our example, simplify \(6-1\).
• Exponents: Next, handle any exponents. Calculate \(5^2\) which equals 25.
• Multiplication and Division: Proceed with multiplication and/or division from left to right. Here, multiply 2 by 25, giving 50.

Following the order of operations is essential for arriving at the correct solution, as performing operations in the wrong order can lead to incorrect answers. This methodical approach helped us correctly evaluate our expression to find the final result of 50.