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Chapter 5: Problem 7

Does 1 raised to any power always equal 1? Why or why not?

Short Answer

Expert verified
Yes, 1 to any power always equals 1 because multiplying 1 by itself any number of times results in 1.

Step by step solution

01

Understanding Exponents

When a number is raised to a power, it means that the base number is multiplied by itself a certain number of times. For example, if we have a base of 3 and an exponent of 2, that's calculated as 3 multiplied by itself once (3 \( \times \) 3), equaling 9. The exponent dictates how many times to multiply the base by itself.
02

Applying Exponents to 1

Let's apply what we understood about exponents to the base number 1. When 1 is raised to any exponent, say \( n \), it is calculated as 1 multiplied by itself \( n \) times, or \( 1^n = 1 \times 1 \times \, ... \, \times 1 \) (a total of \( n \) ones).
03

Realizing Constant Nature of 1

Observe that multiplying 1 by itself any number of times (from 1 to infinity) always results in 1. This is because multiplying by 1 doesn't change the value of the number it's multiplied by. Therefore, for any non-negative integer \( n \), \( 1^n = 1 \) holds true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power of a Number
The concept of the "Power of a Number" involves multiplying a number by itself a certain number of times. This multiplication process is what we call raising a number to a power. Think of it as repeating multiplication.
An example can be seen in the expression \(3^2\). Here, 3 is multiplied by itself once (\(3 \times 3\)), resulting in 9. The expression \(3^2\) highlights the "power" as 2, meaning two occurrences of 3 in multiplication.
An important rule to remember is: the larger the exponent, the more times you multiply the base by itself. This idea helps simplify complex calculations and makes it easy to manage multiple repetitions quickly.
  • The base is the number being multiplied.
  • The exponent tells you how many times to multiply it.
Base and Exponent
Understanding the roles of the base and exponent is key to mastering exponents. The base is the number you want to multiply.
The exponent tells you how many times to multiply it. This creates a simplified way to denote repeated multiplication.
Take \(5^3\) as an example:
  • Base (5): The number being multiplied.
  • Exponent (3): Indicates 5 should be multiplied by itself two more times.
If calculated, \(5^3 = 5 \times 5 \times 5 = 125\). Understanding this pattern makes it easier to tackle more complex problems involving exponents, as these two components determine the calculation's outcome.
Properties of Exponents
The properties of exponents simplify working with repeated multiplication. They are designed to make calculations straightforward and consistent.
  • Power of One: Any number raised to the power of one is itself. For example, \(a^1 = a\).
  • Zero Exponent: Any non-zero number raised to the power of zero equals one: \(a^0 = 1\).
  • Multiplying Powers with the Same Base: Add the exponents: \(a^m \times a^n = a^{m+n}\).
  • Power of a Power: Multiply the exponents: \((a^m)^n = a^{m \times n}\).
These rules help simplify complex expressions and make calculations quicker. Importantly, they follow consistent patterns, making them predictable and easy to use in different contexts. Understanding and applying these properties can solve many problems efficiently.

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