Problem 52 - Algebra 1

Question

Evaluate the exponential expression. \((b-c)^{2}\) when \(b=2\) and \(c=1\)

Step-by-Step Solution

Answer

The evaluated result of the exponential expression \((2-1)^{2}\) is 1.

Step 1: Substitute the given values

First, identify the given values and substitute them into the expression. So \((b-c)^{2}\) becomes \((2-1)^{2}\).

Step 2: Execute Subtraction

After substituting the values, execute the subtraction operation within the brackets. So \((2-1)^{2}\) simplifies to \(1^{2}\).

Step 3: Evaluate Exponential Expression

Finally, evaluate the power of 2, which simply means multiplying 1 by itself. So, \(1^{2}\) equals 1.

Key Concepts

Substitution in Algebra

In algebra, substitution is the process of replacing variables with their given values. This is a crucial first step when working with algebraic expressions. To substitute effectively:

• Identify the variables and their corresponding values provided in the problem.
• Replace each variable in the expression with its given value.

For example, in the expression \((b-c)^{2}\), when you are given that \(b=2\) and \(c=1\), you substitute these values into the expression resulting in \((2-1)^{2}\). This creates a numerical expression that you can then evaluate with basic arithmetic operations. Substitution makes complex algebraic phrases simpler and more manageable as it turns abstract letters into concrete numbers.

Order of Operations

The order of operations is a fundamental concept in mathematics that dictates the correct sequence to evaluate a mathematical expression. It ensures that everyone solves problems in a consistent way. The acronym PEMDAS helps remember the sequence:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the expression \((2-1)^{2}\), you first deal with the parentheses by performing the subtraction: \(2-1 = 1\). After evaluating operations inside the parentheses, you move on to exponents if there are any, as seen in the step-by-step solution where \(1^2\) is next. Following this order prevents mistakes such as miscalculating due to incorrect operational sequence.

Exponential Expressions

Exponential expressions involve numbers raised to a power, indicating how many times the base number is multiplied by itself. In our original problem, after substituting and carrying out the subtraction inside the parentheses, we ended up with \(1^{2}\).

Here:

• The base is 1.
• The exponent is 2.

To evaluate \(1^{2}\), you multiply 1 by itself: \[1 \times 1 = 1\]The beauty of exponential expressions is that they simplify repeated multiplication into a concise form. Understanding how to interpret and evaluate these expressions is essential for tackling more complex math problems that involve powers and exponents.