Applying the Zero Exponent Rule to the Given Expressions

Oct 13, 2025 by magicalceylontours.com 71 views

Introduction

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. However, when dealing with exponents, it's essential to understand the rules and properties that govern their behavior. In this article, we will explore the commonality among four exponential expressions: $4^0$ , $(-2)^0$ , $\left(\frac{1}{3}\right)^0$ , and $9^0$ . We will examine the properties of exponents and determine which statement accurately describes what these powers have in common.

The Zero Exponent Rule

The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. This rule can be expressed mathematically as:

$a^0 = 1$ , where $a \neq 0$

This rule applies to all non-zero numbers, regardless of their magnitude or sign. For example, $2^0 = 1$ , $(-3)^0 = 1$ , and $\left(\frac{1}{2}\right)^0 = 1$ .

Applying the Zero Exponent Rule to the Given Expressions

Now, let's apply the zero exponent rule to the given expressions:

$4^0 = 1$
$(-2)^0 = 1$
$\left(\frac{1}{3}\right)^0 = 1$
$9^0 = 1$

As we can see, all four expressions evaluate to 1, regardless of the base value.

Evaluating the Statements

Now that we have applied the zero exponent rule to the given expressions, let's evaluate the two statements:

A. All the powers have a value of 0 because the exponent is zero.

B. All the powers have a value of 1 because the exponent is zero.

Based on our analysis, statement B is the correct answer. All four expressions have a value of 1 because the exponent is zero.

Conclusion

In conclusion, the zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. When we apply this rule to the given expressions, we find that all four expressions evaluate to 1. Therefore, the correct statement is that all the powers have a value of 1 because the exponent is zero.

Frequently Asked Questions

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1.

Q: Does the zero exponent rule apply to all numbers?

A: Yes, the zero exponent rule applies to all non-zero numbers, regardless of their magnitude or sign.

Q: What is the value of $4^0$ ?

A: The value of $4^0$ is 1.

Q: What is the value of $(-2)^0$ ?

A: The value of $(-2)^0$ is 1.

Q: What is the value of $\left(\frac{1}{3}\right)^0$ ?

A: The value of $\left(\frac{1}{3}\right)^0$ is 1.

Q: What is the value of $9^0$ ?

A: The value of $9^0$ is 1.

Final Thoughts

In this article, we explored the commonality among four exponential expressions: $4^0$ , $(-2)^0$ , $\left(\frac{1}{3}\right)^0$ , and $9^0$ . We applied the zero exponent rule to each expression and found that all four expressions evaluate to 1. Therefore, the correct statement is that all the powers have a value of 1 because the exponent is zero. We hope this article has provided a clear understanding of the zero exponent rule and its application to exponential expressions.

Introduction

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. However, when dealing with exponents, it's essential to understand the rules and properties that govern their behavior. In this article, we will answer some frequently asked questions about exponents, covering topics such as the zero exponent rule, negative exponents, and more.

Q&A: Exponents

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. This rule can be expressed mathematically as:

$a^0 = 1$ , where $a \neq 0$

Q: Does the zero exponent rule apply to all numbers?

A: Yes, the zero exponent rule applies to all non-zero numbers, regardless of their magnitude or sign.

Q: What is the value of $4^0$ ?

A: The value of $4^0$ is 1.

Q: What is the value of $(-2)^0$ ?

A: The value of $(-2)^0$ is 1.

Q: What is the value of $\left(\frac{1}{3}\right)^0$ ?

A: The value of $\left(\frac{1}{3}\right)^0$ is 1.

Q: What is the value of $9^0$ ?

A: The value of $9^0$ is 1.

Q: What is the rule for negative exponents?

A: The rule for negative exponents states that any non-zero number raised to a negative power is equal to the reciprocal of the number raised to the positive power. This rule can be expressed mathematically as:

$a^{-n} = \frac{1}{a^n}$ , where $a \neq 0$ and $n$ is a positive integer.

Q: What is the value of $4^{-2}$ ?

A: The value of $4^{-2}$ is $\frac{1}{4^2} = \frac{1}{16}$ .

Q: What is the value of $(-2)^{-3}$ ?

A: The value of $(-2)^{-3}$ is $\frac{1}{(-2)^3} = \frac{1}{-8} = -\frac{1}{8}$ .

Q: What is the rule for multiplying exponents with the same base?

A: The rule for multiplying exponents with the same base states that when multiplying two numbers with the same base, you add the exponents. This rule can be expressed mathematically as:

$a^m \cdot a^n = a^{m+n}$ , where $a$ is a non-zero number and $m$ and $n$ are integers.

Q: What is the value of $2^3 \cdot 2^4$ ?

A: The value of $2^3 \cdot 2^4$ is $2^{3+4} = 2^7 = 128$ .

Q: What is the rule for dividing exponents with the same base?

A: The rule for dividing exponents with the same base states that when dividing two numbers with the same base, you subtract the exponents. This rule can be expressed mathematically as:

$\frac{a^m}{a^n} = a^{m-n}$ , where $a$ is a non-zero number and $m$ and $n$ are integers.

Q: What is the value of $\frac{2^5}{2^3}$ ?

A: The value of $\frac{2^5}{2^3}$ is $2^{5-3} = 2^2 = 4$ .

Conclusion

In this article, we have answered some frequently asked questions about exponents, covering topics such as the zero exponent rule, negative exponents, and more. We hope this article has provided a clear understanding of the rules and properties of exponents and has helped to clarify any confusion.

Final Thoughts

Exponents are a fundamental concept in mathematics, and understanding the rules and properties that govern their behavior is essential for success in mathematics and science. We hope this article has been helpful in providing a clear understanding of exponents and has inspired you to learn more about this fascinating topic.

Additional Resources

For more information on exponents, we recommend the following resources:

Khan Academy: Exponents
Mathway: Exponents
Wolfram MathWorld: Exponents

We hope this article has been helpful in providing a clear understanding of exponents and has inspired you to learn more about this fascinating topic.