Problem 54 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(25^{3}\)

Step-by-Step Solution

Answer

The value of \(25^{3}\) is 15625.

Step 1: Understand the Notation

The expression \(25^{3}\) means 25 raised to the power of 3. This implies that the base number 25 is multiplied by itself three times.

Step 2: Compute the Power Manually

To manually compute \(25^{3}\), multiply 25 by itself three times: \(25 \times 25 \times 25\). First, multiply 25 by 25 to get 625: \(25 \times 25 = 625\). Then, multiply the result by 25 again: \(625 \times 25 = 15625\).

Step 3: Verify with a Calculator

To ensure accuracy, use a calculator to compute \(25^{3}\). Enter 25, then press the exponent key (usually labeled with a '^' or similar icon), then enter 3, and press equal. The calculator should confirm that the answer is 15625.

Key Concepts

Understanding Powers

A power is a mathematical expression involving two numbers: a base and an exponent. When we talk about powers, we're generally referring to the process of exponentiation. This involves raising a number (the base) to the power of another number (the exponent). In the expression \(25^{3}\), 25 is the base and 3 is the exponent.

Let's break it down:

• The base number (25) tells us what number we're repeatedly multiplying.
• The exponent (3) tells us how many times to multiply the base by itself.

So, \(25^{3}\) means 25 multiplied by 25, and then that result multiplied by 25 again. Essentially, the process is:

• Start with one instance of the base (25).
• Multiply it by 25 once more to get 625.
• Then multiply the result (625) by 25 to get 15625.

This is why it's important to understand the notation and meaning behind powers, as it simplifies working through and solving exponential expressions. Understanding this can also help in communicating mathematical ideas more clearly.

The Role of Multiplication

Multiplication is a key operation in mathematics that enables us to find the product of two numbers. When we deal with powers, we are essentially utilizing repeated multiplication. In the expression \(25^{3}\), the operation is taking 25, and multiplying it three times.

Here's how multiplication acts:

• First, multiply 25 by itself to get 625.
• Next, multiply the result (625) by 25 to arrive at 15625.

Each step requires basic multiplication skills. It's important to do these steps carefully to ensure the results are accurate. The magic of multiplication in exponents is that it enables us to efficiently calculate large numbers quickly, making tasks that would otherwise take a long time much easier.

Furthermore, mastering multiplication provides a strong foundation for understanding more complex mathematical concepts, including other algebraic operations and solving equations.

Calculator Usage for Powers

Calculators are powerful tools for solving math problems efficiently, especially when dealing with large calculations like powers. Once you're familiar with the math underlying exponents, using a calculator can help verify results quickly.

When inputting \(25^{3}\) into a calculator, follow these steps to ensure you enter the expression correctly:

• Enter 25, the base number.
• Find and press the exponent function key, often labeled as '^' or 'EXP'.
• Enter 3, the exponent.
• Press the equal button to compute the result, which should display 15625.

Using a calculator reduces the risk of error, especially with larger exponents. It is essential to double-check your manual calculations with a calculator to confirm accuracy. Also, practicing calculator usage can improve efficiency and are beneficial for tasks requiring quick computation.