Problem 85 - Prealgebra

Question

Expand and multiply. $$3^{2}$$

Step-by-Step Solution

Answer

The expanded and multiplied result of \(3^2\) is 9.

Step 1: Identify Base and Exponent

The expression given is \(3^2\). Here, the base is 3, and the exponent is 2. This tells us we need to multiply the base by itself as many times as the exponent indicates.

Step 2: Multiply the Base

Since the exponent is 2, we multiply the base (3) by itself: \(3 \times 3\).

Step 3: Calculate the Product

Perform the multiplication from Step 2. \(3 \times 3 = 9\). This result is the expanded and multiplied value of \(3^2\).

Key Concepts

Understanding Base and Exponent

When dealing with exponents, recognizing the base and exponent in an expression is crucial. Consider the expression \(3^2\). Here, the base is the number 3, and the exponent is 2. The base is the number that you will be multiplying repeatedly, and the exponent tells you how many times to multiply the base by itself. It is a way of expressing repeated multiplication easily, using just a small, superscript number (the exponent) placed next to the base.

• The base: The number you see before the exponent.
• The exponent: The small, raised number to the upper-right of the base.

For example, in \(3^2\), you multiply 3 by itself because the exponent is 2. The concept of base and exponent simplifies expressions, allowing for more neat and compact equations.

The Role of Multiplication

Multiplication is at the heart of understanding exponents. Once you have identified the base and exponent, you use multiplication to expand the expression. In \(3^2\), multiplication is your tool to derive the final value. This is because the exponent 2 indicates that you should multiply the base (3) by itself. Remember these key points when multiplying:

• Multiplication helps in scaling a number by combining several of the same number.
• With an exponent, multiplication occurs between the same numbers based on the exponent value.

For \(3^2\), you multiply the base, 3, by itself exactly once (as shown by the exponent). So, you calculate \(3 \times 3\). Understanding how multiplication works with exponents will aid in solving more complex problems down the line.

Calculating Powers

Once you have the setup with the base and multiplication, it's time to calculate the power. "Calculating powers" is just another way of saying, "solve the multiplication." In \(3^2\), after setting up the multiplication as \(3 \times 3\), you perform the multiplication to get the final result. Here’s the breakdown:

• Take the base, which is 3.
• Multiply 3 by itself once since the exponent is 2, which simplifies to \(3 \times 3\).
• The result is 9.

This result is the power, i.e., \(3\) raised to the power of \(2\), gives us 9. The calculated power is a concise and efficient way to represent heavily repetitive multiplication, making math both easier and more intuitive.