Problem 48 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression evaluates to 50.

Step 1: Substitute Values into the Expression

Take the given values of the variables and substitute them into the expression. The expression is \(2(a+b)^2\) and we have \(a=6\) and \(b=-1\). Replace \(a\) with 6 and \(b\) with -1.

Step 2: Simplify Inside the Parentheses

After substitution, the expression inside the parentheses is \((6 + (-1))\). Compute this addition: \(6 - 1 = 5\).

Step 3: Apply Squaring Operation

Take the result from Step 2 and square it. We have \(5^2 = 25\).

Step 4: Multiply by the Constant

Now multiply the squared result by the constant factor outside the parentheses. Compute \(2 \times 25 = 50\).

Key Concepts

Substitution in Algebra

Substitution in algebra is the process of replacing variables in an expression with their given numerical values. It is an essential skill that helps simplify expressions and solve equations. Imagine an algebraic expression as a recipe, and the variables are the ingredients. When you know the quantity for each ingredient, you can successfully substitute them, just like in a recipe, to create a finished dish.

• Start by identifying the variables in your expression and their corresponding values.
• Substitute these values wherever the variables appear in the expression.

In our example, the expression is \(2(a+b)^2\), where \(a=6\) and \(b=-1\). By substituting, we replace \(a\) with 6 and \(b\) with -1 to get \(2(6+(-1))^2\). This step sets the stage for the arithmetic operations that follow.

Simplifying Algebraic Expressions

Once substitution is done, the next crucial step is simplifying the algebraic expression. Simplifying involves performing the arithmetic operations as indicated and reducing the expression to its most straightforward form.

• Focus on solving the operations inside parentheses first.
• Then handle the exponents or any multiplications/divisions as necessary.

In the provided exercise, after substituting the values, the expression becomes \(2(6 + (-1))^2\). Simplify the expression inside the parentheses: \((6 - 1 = 5)\).

Next, apply the exponent by squaring 5, which gives us 25. At this point, the expression is now \(2 \times 25\). Simplifying helps us proceed efficiently to the solution without unnecessary steps.

Order of Operations in Algebra

Order of operations is a fundamental principle in algebra that ensures consistent and correct results. It is critical to follow a specific order when solving expressions to avoid errors.

• First, solve any expressions inside parentheses or brackets.
• Next, address exponents or powers.
• After dealing with exponents, move on to multiplication and division.
• Finally, handle any addition and subtraction.

In our example, the order was to evaluate \((6 + (-1))\), then square the result to get 25, and lastly, multiply it by 2, resulting in 50.

By adhering to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), you ensure that expressions are evaluated correctly and efficiently.