Problem 57 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(15^{5}\)

Step-by-Step Solution

Answer

The value of \(15^5\) is 759375.

Step 1: Understanding the Problem

We need to find the value of the expression given by the power \(15^5\). This means we have to multiply the base number 15 by itself a total of 5 times.

Step 2: Calculate the Power Manually

To calculate \(15^5\), we multiply the base by itself:\[15 \times 15 \times 15 \times 15 \times 15\]We can do this sequentially. First calculate \(15^2 = 225\), then \(225 \times 15 = 3375\), and continue multiplying by 15: \( 3375 \times 15 = 50625\), \( 50625 \times 15 = 759375\).

Step 3: Verify with a Calculator

Now use a calculator to check your manual computation: Enter 15 raised to the power of 5 (often done by punching in 15, then the exponent key, and finally the number 5). The calculator should read \(15^5 = 759375\), confirming our manual calculation.

Key Concepts

Understanding Powers in Mathematics

When dealing with powers or exponentiation, we are essentially talking about a process of repeated multiplication. A power is written in the form \(a^n\), where \(a\) is the base number and \(n\) is the exponent, or power. This notation tells us to multiply the base number \(a\) by itself \(n\) times. For example, with \(15^5\), the base 15 is multiplied by itself five times.

• It simplifies how we express large multiplications.
• Makes calculations manageable and organized.
• Provides a foundational concept in algebra and higher mathematics.

Understanding powers helps us discover how numbers can grow rapidly and predictably when multiplied multiple times.

Identifying the Base Number

In any expression involving powers, identifying the base number is crucial. The base number is the number that gets repeatedly multiplied. In the expression \(15^5\), the base is clearly 15. This number serves as the building block of the operation. Having a strong grasp of what the base is helps to correctly interpret and solve problems using exponentiation.

• The base determines the starting point of the multiplication.
• It is always the number that appears before the exponent.
• In simple terms, it's the 'thing' being multiplied several times.

Knowing the base allows you to correctly perform each step of the exponentiation.

Importance of Calculation Verification

Calculation verification ensures accuracy, which is particularly essential when working with powers, as errors can easily arise from manual computation. After determining the result of a power through sequential multiplication, verifying this result with a calculator helps confirm its correctness.
Using a calculator involves entering the base number followed by the exponent key and the exponent value. If both methods return the same result, as with the example \(15^5 = 759375\), it reinforces the accuracy of the solution.

• Helps identify calculation errors.
• Combines manual and technological methods for data validation.
• Promotes confidence in mathematical problem-solving.

This process is an essential check that balances manual calculation intricacies with computational power.

Sequential Multiplication in Finding Powers

Sequential multiplication is the process of breaking down the multiplication of a base number by itself into simpler steps. This method helps manage large calculations by reducing them to smaller, more manageable parts. When calculating \(15^5\), you can first compute \(15^2\), then use that result to multiply again by 15, and so forth.
For example:

• Compute \(15^2 = 225\).
• Multiply \(225 \times 15 = 3375\).
• Multiply \(3375 \times 15 = 50625\).
• Finally, multiply \(50625 \times 15 = 759375\).

Sequential multiplication not only organizes the steps but also reduces the cognitive load associated with handling large numbers directly. It guides you through each necessary multiplication step and aids in maintaining accuracy throughout the calculation.