Problem 46 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(6^{2}\)

Step-by-Step Solution

Answer

The value of \(6^2\) is 36.

Step 1: Understand the Notation

The notation \(6^2\) means "6 squared" or "6 raised to the power of 2." This indicates that the number 6 is multiplied by itself.

Step 2: Perform the Multiplication

Calculate \(6^2\) by multiplying 6 by itself. This is equivalent to \(6 \times 6\).

Step 3: Calculation

Perform the multiplication: \(6 \times 6 = 36\).

Step 4: Validate with a Calculator

Use a calculator to confirm the calculation of the exponent. Input \(6^2\) into the calculator to ensure the result is 36.

Key Concepts

Power Notation

Understanding power notation is essential when working with exponents. In mathematics, the expression \(6^2\) is an example of power notation. Here, the base number is 6, and the exponent is 2. This is often read as "6 squared." Power notation is a concise way to represent repeated multiplication of a number by itself.
For instance:

• \(6^2\) means multiply 6 by itself once, which is 6 times 6.
• \(6^3\) would mean multiply 6 by itself twice, resulting in 6 times 6 times 6.

You can think of the exponent as a shortcut that tells you how many times to multiply the base by itself.
It simplifies repeated calculations and makes it easier to communicate large multiplications.

Multiplication

Once you've understood the basics of power notation, the next step involves performing the actual multiplication. Using the example \(6^2\), you multiply 6 by itself: 6 times 6. In this way, power notation directly leads to multiplication.
With multiplication of exponents:

• \(6 \times 6 = 36\), the answer to \(6^2\).

By understanding that the exponent indicates how many times the base should be multiplied together, you can solve for powers quickly.
This multiplication is straightforward, especially with smaller numbers, allowing you to easily manage and calculate values by hand or in your head.

Calculator Validation

After you've computed your answer using power notation and multiplication, it's always a good practice to double-check your results. Calculator validation ensures accuracy. To validate using a calculator, simply input the expression \(6^2\) directly.
Here's how you can verify it:

• Enter 6, then press the exponent function (often labeled as ^ or xy), followed by 2.
• Observe the displayed result, which should be 36.

Using a calculator reduces errors and saves time, especially with more complex or large numbers. By validating your calculations, you gain confidence in your answer and enhance your problem-solving skills.