Problem 35 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(12^{2}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Base and Exponent

In the expression \(12^{2}\), identify the base and the exponent. The base is 12, and the exponent is 2. This means you are multiplying 12 by itself once.

Step 2: Calculate the Power

To calculate \(12^{2}\), multiply the base by itself as many times as indicated by the exponent. Here, you'll calculate \(12 \times 12\).

Step 3: Perform the Multiplication

Calculate \(12 \times 12\). The result of this multiplication is 144.

Step 4: Verify Using a Calculator

Enter the expression \(12^{2}\) into a calculator to verify the result. The calculator should also show the result as 144.

Step 5: Confirm the Solution

Both the manual calculation and the verification step should agree, confirming that \(12^{2} = 144\).

Key Concepts

Understanding Base and Exponent

In mathematics, especially when dealing with exponents, it is crucial to grasp the concept of the base and exponent. The base is the number that is subject to repeated multiplication, while the exponent tells us how many times the base number is multiplied by itself. For example, in the expression \( 12^2 \), 12 is the base, and 2 is the exponent.
When interpreting this, the exponent signifies that the base should be used as a factor twice. This is a foundational concept in exponents, helping translate exponential expressions into a form that's easier to compute.

Performing Power Calculation

Once we have identified the base and exponent, the next task is to perform the power calculation. This involves multiplying the base by itself according to the number of times specified by the exponent.
Let's look at the expression \( 12^2 \). The exponent is 2, which means you multiply 12 by 12. Mathematically, you execute: \( 12 \times 12 \). This is the process of calculating the power, which simplifies calculating large numbers without writing extensive multiplication.

Carrying Out Multiplication Verification

After performing a power calculation, it's important to verify that your result is correct. This can be done by using a calculator or re-evaluating the multiplication. Verification serves as an essential step to ensure accuracy in mathematical solutions.
For the calculation \( 12^2 \), you can enter this expression into a calculator to confirm that the result of \( 12 \times 12 \) is indeed 144. The agreement between the manual calculation and the calculator's outcome confirms the correctness of the solution. Regular practice of verification helps build confidence and accuracy in dealing with powers and exponents.