Problem 48 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(1^{28}\)

Step-by-Step Solution

Answer

The value of \(1^{28}\) is 1.

Step 1: Identify the Base and Exponent

The given expression is \(1^{28}\), where 1 is the base and 28 is the exponent. An exponent signifies how many times to multiply the base by itself.

Step 2: Apply the Power Rule for One

When any non-zero number is raised to the power of zero, the result is 1. However, when 1 is raised to any power, like in \(1^{28}\), the result remains 1. This is because multiplying 1 by itself any number of times still gives a product of 1.

Step 3: Verify with a Calculator

To confirm, you can use a calculator to evaluate \(1^{28}\). Enter the base 1, press the exponentiation key, and then enter 28. The display should show 1, confirming that \(1^{28} = 1\).

Key Concepts

Power Rule

The power rule refers to the shortcut we use when dealing with exponents, especially when the base is consistent across calculations. The rule simplifies complex calculations and helps in readily understanding exponent behavior. It states that when a number is raised to another number, called the exponent, it's equivalent to multiplying the base by itself that many times. For instance,

• In \( 1^{28} \), the base 1, is repeated 28 times.
• Since 1 multiplied by itself any number of times is always 1, this simplification always results in 1 when 1 is the base.

The rule most commonly noted alongside basics of exponentiation is that any number raised to the power of 0 equals 1, except when that number is 0.

Base and Exponent

In mathematics, understanding the terms "base" and "exponent" is crucial for mastering exponents. The base is the number you are multiplying, and the exponent tells you how many times to multiply that number by itself. For example, in the expression \( 1^{28} \):

• The number 1 is the base.
• The number 28 is the exponent.

This means that you multiply the base, 1, by itself 28 times.
Yet, because multiplying 1 by itself is always 1, it simplifies the solution to 1. This highlights the unique properties of using 1 as a base, which can be one of the simplest calculations involving exponents.

Using a Calculator

Using a calculator to evaluate expressions with exponents can be very helpful, especially for checking your work. Calculators are designed to handle large numbers efficiently.

• First, you input the base, which is 1 in our case.
• Next, press the exponentiation function, often marked as a caret '^' or a special key for powers.
• Then, enter the exponent, which is 28.

The calculator will compute and display the result. For this expression, \( 1^{28} \), the calculator should return 1.
While this example is straightforward, learning to use a calculator for exponents builds confidence for more complex calculations involving different numbers.