Problem 47 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(7^{1}\)

Step-by-Step Solution

Answer

The value of \(7^1\) is 7.

Step 1: Identify the Base and Exponent

In the expression \(7^1\), the base is 7 and the exponent is 1. This means 7 is raised to the power of 1.

Step 2: Calculate the Power

To find \(7^1\), recall that any number raised to the power of 1 is the number itself. Therefore, \(7^1 = 7\).

Step 3: Verify with a Calculator

Use a calculator to enter the base (7) and the exponent (1). The calculator should confirm that \(7^1 = 7\).

Key Concepts

Base and Exponent

In expressions involving exponents, two components play crucial roles: the base and the exponent itself. The base is the number that is being multiplied by itself, and the exponent indicates how many times the base is used as a factor.
For example, in the expression \(7^1\), the base is 7. The exponent, in this case, is 1.

• The base (7) tells us the number we are working with.
• The exponent (1) shows that we use the base just once.

This expression simplifies to adapting the base as it is because any number raised to the power of 1 remains unchanged. This fundamental rule simplifies calculations and helps with understanding more complex exponents.

Powers of Numbers

Understanding powers of numbers is fundamental to mastering exponents. When a number is raised to a power, it implies repeated multiplication of that number. For instance, \(7^1\) means 7 multiplied once or simply 7 itself.
But what really happens when the exponent changes?

• When the exponent is 2 (as in \(7^2\)), it means 7 is multiplied by itself: \(7 \times 7\).
• For an exponent of 3 (\(7^3\)), the calculation would be \(7 \times 7 \times 7\).
• Generally, any number to the power of 1 is always the number itself.

Recognizing these patterns helps us comprehend how exponents affect numbers and how to compute them more effectively.

Calculator Verification

Calculators are valuable tools for confirming calculations, especially when dealing with exponents. Although straightforward calculations like \(7^1\) can be done manually, calculator verification is useful to ensure accuracy. It also comes in handy for more complex operations.

• Enter the base number (7) into the calculator.
• Use the exponent function to specify the power (1).
• Observe that the calculator indeed shows \(7^1\) equals 7.

This simple verification step ensures that calculations are carried out correctly. It provides a safety net for understanding complex expressions and cements confidence in using mathematical tools efficiently.