Problem 37 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(15^{2}\)

Step-by-Step Solution

Answer

The value of \(15^2\) is 225.

Step 1: Identify the Base and Exponent

In the expression \(15^2\), 15 is the base and 2 is the exponent. This means we will multiply the base, 15, by itself, 2 times.

Step 2: Multiply the Base by Itself

Since the exponent is 2, multiply 15 by itself: \(15 \times 15\).

Step 3: Calculate the Product

Perform the multiplication: \(15 \times 15 = 225\).

Step 4: Verify Using a Calculator

Use a calculator to ensure the multiplication was done correctly. Enter 15, use the exponentiation function to input the exponent 2, and confirm that the result is indeed 225.

Key Concepts

Understanding Base and Exponent

In mathematics, the terms "base" and "exponent" are foundational to understanding powers. When you see an expression like \(15^2\), the base is the number 15, indicating what number you are multiplying. The exponent, or the little number written above and to the right (2 in this case), tells how many times you multiply the base by itself.
To visualize this, think of the base as the starting block, and the exponent tells you how many more blocks you need. Each block or step in this visualization represents one multiplication of the base.
Simply put:

• The base (15 in our example) is the number being multiplied.
• The exponent (2) is the number of times the base appears in the multiplication.

The Process of Multiplication with Exponents

Multiplication is a fundamental operation in mathematics, and when combined with exponents, it simplifies repeated multiplication into a more compact form. With the expression \(15^2\), multiplication is performed by repeatedly multiplying the base, 15, by itself.
Here's a step-by-step breakdown:

• Start with the base: 15.
• Since the exponent is 2, multiply 15 by itself: \(15 \times 15\).
• This process results in the product of 225, the final answer.

Multiplication like this reduces long chains of arithmetic into more manageable operations. Familiarizing yourself with multiplication tables or performing manual calculations builds a strong foundation for understanding more complex math problems.

Calculator Use for Verification

Using a calculator is an excellent way to confirm the accuracy of your mathematical calculations, ensuring you have the correct result. Most modern calculators have an exponentiation function, which simplifies the process of calculating powers.
To verify \(15^2\):

• Enter the base number (15) into the calculator.
• Use the exponentiation function, usually denoted as \(^\), \(x^y\), or \(\text{exp}\).
• Enter the exponent (2).
• Press "equals" or "enter" to see the result.
• The calculator should show 225, confirming your manual calculation.

This method is not just for verification but also teaches how powerful calculators can be as tools in simplifying and checking work.

Mathematical Verification and Solid Understanding

Mathematical verification is the step where you ensure calculations were done correctly, providing peace of mind that the answer is accurate.
After performing a manual calculation of \(15^2\) and using a calculator to confirm, you've practiced verification, which reinforces your understanding.Verification isn't just about finding errors; it's about:

• Reinforcing the process of multiplication and exponentiation.
• Building confidence in mathematical skills.
• Ensuring comprehension of the operations performed.

By routinely verifying answers, you also develop a habit of critical thinking and problem-solving, which are crucial in more advanced mathematical studies.