Problem 104 - Intermediate Algebra

Question

Evaluate each expression. $$ \text { a. } 2 \cdot 3^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to 18.

Step 1: Identify the Base and the Exponent

In the expression \(2 \cdot 3^{2}\), observe that \(3^{2}\) is the part with a base and an exponent. The base is 3, and the exponent is 2. This means you will be multiplying 3 by itself.

Step 2: Calculate the Power

To solve \(3^{2}\), multiply 3 by itself. \(3 \times 3 = 9\). Now, replace \(3^{2}\) in the expression with 9.

Step 3: Multiply the Result by the Coefficient

Substitute the result from the power into the expression: \(2 \cdot 9\). Now, multiply 2 by 9 to find the final result.

Step 4: Calculate the Final Result

Perform the multiplication: \(2 \times 9 = 18\). This is the value of the expression.

Key Concepts

Base and Exponent

Understanding the concepts of base and exponent is essential in working with powers in mathematics. In the expression \(3^2\), 3 is known as the _base_, and it is the number that is being multiplied. The number 2 is the _exponent_, which tells us how many times to multiply the base by itself.

• The base is the number we want to multiply.
• The exponent indicates the number of times the base is used as a factor.

This means that the exponent guides us on the number of times the base number should be multiplied together. In simple terms, \(3^2\) means you will multiply 3 by itself, resulting in 3 times 3. The focus should be on recognizing the distinct roles of bases and exponents to avoid confusion in performing calculations correctly.

Apply Order of Operations

When evaluating mathematical expressions like \(2 \cdot 3^2\), it's crucial to apply the correct sequence of operations to arrive at the correct answer. This is dictated by 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)).

• Step 1: Take care of any operations within parentheses first.
• Step 2: Evaluate powers and roots, i.e., exponents.
• Step 3: Proceed with multiplication and division from left to right.
• Step 4: Finally, handle addition and subtraction from left to right.

For the expression \(2 \cdot 3^2\), we start with the exponent (\(3^2\)), process it, and only then do we move to multiplying the results with 2. This ensures the calculations maintain mathematical integrity and accuracy.

Evaluate Expressions

Evaluating an expression involves performing the arithmetic operations in the correct order as defined by mathematical principles. After identifying the base and exponent and applying the order of operations, you can correctly evaluate the expression. Let's look at how this process unfolds using \(2 \cdot 3^2\) as an example.

• First, solve the exponent to simplify it: \(3^2 = 9\).
• Replace \(3^2\) in the original expression with the calculated result: \(2 \cdot 9\).
• Multiply the numbers: \(2 \times 9 = 18\).

By following these steps, you can systematically and accurately find the value of expressions. Practicing these methods improves your ability to tackle even more complex mathematical problems with ease.