Problem 5 - Fundamentals of Mathematics

Question

Determine the value of each power and root. \(12^{2}\)

Step-by-Step Solution

Answer

The value of \(12^2\) is 144.

Step 1: Identify the Base and Exponent

In the expression \(12^2\), the base is \(12\) and the exponent is \(2\). This means that we will need to multiply the base, \(12\), by itself as many times as indicated by the exponent, which is \(2\).

Step 2: Multiply the Base by Itself

Compute \(12 \times 12\) to determine the value of \(12^2\). This step requires carrying out a simple multiplication.

Step 3: Calculate the Multiplication

Perform the multiplication: \(12 \times 12 = 144\). So, the value of \(12^2\) is \(144\).

Key Concepts

Understanding Powers

When we talk about powers in mathematics, we are dealing with expressions that have a base number and an exponent. A power is represented as a small number, the exponent, written above and to the right of a larger number, the base. The power indicates how many times the base number is multiplied by itself. For example:

• In the expression \(12^2\), the base is 12 and the exponent is 2.
• This tells us that we multiply 12 by itself, two times.

Using powers allows us to write repeated multiplication of the same number in a compact way. Instead of writing \(12 \times 12\), we can efficiently express it using powers as \(12^2\).
Powers are fundamental in various areas of math and science, providing ease in calculations and expressing large numbers succinctly.

Exploring Base and Exponent

Now that we know powers include a base and an exponent, let's dive deeper into what these components mean.
The **base**, in our example, is 12. The base is the number that gets multiplied. It can be any integer, fraction, or decimal, depending on the context of the problem.

The **exponent**, in this case, is 2. It indicates how many times the base is used in a multiplication. Think of the exponent as a shorthand note that tells you to multiply the base by itself a certain number of times.

• If the exponent is 1, you simply get the base back. For example, \(12^1 = 12\).
• If the exponent is 0, any non-zero base raised to the power of zero is 1. Thus, \(12^0 = 1\).
• A negative exponent indicates division or a reciprocal of the positive exponent. For example, \(12^{-1} = \frac{1}{12}\).

Understanding how the base and exponent work together helps in simplifying and solving more complex expressions.

The Role of Multiplication in Powers

Multiplication plays a key role in understanding powers. When you see an exponent, it implies repeated multiplication of the base. In the expression \(12^2\), calculating its value involves multiplying 12 by 12.
Here’s how the process works:

• Identify the operation: With \(12^2\), you recognize that the task is to multiply 12 twice, as indicated by the exponent 2.
• Execute the multiplication: Perform the operation \(12 \times 12\).
• Find the result: Here, 12 multiplied by 12 equals 144.

This simple example highlights how multiplication underpins the calculation of powers. Mastering multiplication makes it easier to handle powers and to solve more intricate math problems effectively.