Problem 118 - Elementary Algebra

Question

Write your own example of an expression that includes one set of parentheses and an exponent. When this expression is evaluated, it equals 4 .

Step-by-Step Solution

Answer

The expression is \( (1 + 1)^2 \).

Step 1: Determine the Structure

To solve the problem, start by determining the structure of the expression. Begin with an expression that includes parentheses and an exponent. An example is \( (a + b)^2 \), where \(a\) and \(b\) are numbers.

Step 2: Decide the Values for a and b

Choose values for \(a\) and \(b\) such that the entire expression evaluates to 4. A simple choice is \(a = 1\) and \(b = 1\). So, the expression becomes \( (1 + 1)^2 \).

Step 3: Perform the Calculation Inside the Parentheses

Calculate the result of the expression inside the parentheses first: \( 1 + 1 = 2 \).

Step 4: Applying the Exponent

Now, apply the exponent to the result from the previous step: \( 2^2 = 4 \).

Step 5: Verify the Result

Finally, ensure that the initial condition is satisfied: \( (1 + 1)^2 = 4 \), which is true.

Key Concepts

Parentheses in Algebra

Using parentheses in algebra is crucial as they determine the order in which operations are performed. Without them, mathematical expressions might look confusing or be interpreted incorrectly. Parentheses indicate that the operations inside should be performed first. This is part of the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
For example, in the expression (1 + 1)^2:

• First, the addition inside the parentheses is executed: 1 + 1 = 2
• Then, the result is taken to the power of 2: 2^2 = 4

Neglecting parentheses can lead to entirely different results. For instance, without the parentheses in the expression 1 + 1^2, you would first apply the exponent, resulting in: 1 + 1 = 2 without the need to square the addition.

Using Exponents

Exponents are a way of representing repeated multiplication of the same number. They are written as a small number (the exponent) to the upper right of a larger number (the base). For example, in the expression 2^2, 2 is the base, and 2 is the exponent, meaning you multiply 2 by itself once: 2^2 = 2 * 2 = 4
It’s important to understand the role of exponents in algebra as they can dramatically change the value of an expression. Here’s a few things to keep in mind:

• Any number raised to the power of 0 is always 1: For example, 5^0 = 1
• Any number raised to the power of 1 is the number itself: For example, 5^1 = 5
• Exponents can also be negative, indicating repeated division: For example, 2^-1 = 1/2

Understanding these rules helps in evaluating more complex algebraic expressions accurately.

Step-by-Step Algebra Problems

Solving algebraic expressions step-by-step not only ensures accuracy but also makes the process easy to follow. When dealing with complex problems, it’s beneficial to break them down into smaller, more manageable steps. Let’s walk through an example:
Suppose we need to evaluate the expression (1 + 1)^2 and show it equals 4.

• Step 1: Determine the Structure First, identify an appropriate structure which includes parentheses and an exponent. In this instance, an example structure is (a + b)^2.
• Step 2: Choose the Values Assign suitable values to a and b. Here, we choose a = 1 and b = 1, transforming our expression to (1 + 1)^2.
• Step 3: Perform the Calculation Inside the Parentheses Calculate the addition inside the parentheses: 1 + 1 = 2.
• Step 4: Apply the Exponent Raise the result to the power of 2: 2^2 = 4.
• Step 5: Verify the Result Confirm that our expression meets the initial condition: (1 + 1)^2 = 4.

By following these steps closely, resolving algebraic expressions becomes more straightforward and less prone to error.