If $c$ Is A Positive Real Number And $m$ And $n$ Are Positive Integers, Then $c^{m / N}=\sqrt[n]{c^m}=(\sqrt[n]{c})^m$.A. True B. False

Feb 27, 2025 by ADMIN 137 views

The Power of Exponents: Understanding the Relationship Between $c^{m / n}$ , $\sqrt[n]{c^m}$ , and $(\sqrt[n]{c})^m$

In mathematics, exponents and roots are fundamental concepts that play a crucial role in algebra and beyond. The relationship between exponents and roots is a fascinating topic that has been extensively studied and explored. In this article, we will delve into the world of exponents and roots, and examine the statement: "If $c$ is a positive real number and $m$ and $n$ are positive integers, then $c^{m / n}=\sqrt[n]{c^m}=(\sqrt[n]{c})^m$ ." We will explore the underlying mathematics, provide examples, and discuss the implications of this statement.

The Basics of Exponents and Roots

Before we dive into the statement, let's review the basics of exponents and roots.

Exponents: An exponent is a small number that is written above and to the right of a base number. It represents the number of times the base number is multiplied by itself. For example, $2^3$ means $2$ multiplied by itself $3$ times, which equals $8$ .
Roots: A root is the inverse operation of an exponent. It represents the number that, when raised to a certain power, equals a given value. For example, $\sqrt{16}$ means the number that, when squared, equals $16$ , which is $4$ .

Understanding the Statement

The statement claims that if $c$ is a positive real number and $m$ and $n$ are positive integers, then $c^{m / n}=\sqrt[n]{c^m}=(\sqrt[n]{c})^m$ . Let's break down each part of the statement:

$c^{m / n}$ : This represents the exponentiation of $c$ to the power of $m/n$ . In other words, it means $c$ multiplied by itself $m/n$ times.
$\sqrt[n]{c^m}$ : This represents the $n$ th root of $c^m$ . In other words, it means the number that, when raised to the power of $n$ , equals $c^m$ .
$(\sqrt[n]{c})^m$ : This represents the exponentiation of the $n$ th root of $c$ to the power of $m$ . In other words, it means the $n$ th root of $c$ multiplied by itself $m$ times.

Proof of the Statement

To prove the statement, we can use the following steps:

Step 1: Start with the expression $c^{m / n}$ .
Step 2: Rewrite $c^{m / n}$ as $(c^{1/n})^m$ using the property of exponents.
Step 3: Recognize that $(c^{1/n})^m$ is equivalent to $\sqrt[n]{c^m}$ .
Step 4: Conclude that $c^{m / n}=\sqrt[n]{c^m}$ .

Example

Let's consider an example to illustrate the statement. Suppose we have $c=2$ , $m=3$ , and $n=2$ . We can calculate the values of $c^{m / n}$ , $\sqrt[n]{c^m}$ , and $(\sqrt[n]{c})^m$ as follows:

$c^{m / n}$ : $2^{3/2} = \sqrt{2^3} = \sqrt{8} = 2\sqrt{2}$
$\sqrt[n]{c^m}$ : $\sqrt{2^3} = \sqrt{8} = 2\sqrt{2}$
$(\sqrt[n]{c})^m$ : $(\sqrt{2})^3 = 2\sqrt{2}$

As we can see, the values of $c^{m / n}$ , $\sqrt[n]{c^m}$ , and $(\sqrt[n]{c})^m$ are all equal, which confirms the statement.

Conclusion

In conclusion, the statement "If $c$ is a positive real number and $m$ and $n$ are positive integers, then $c^{m / n}=\sqrt[n]{c^m}=(\sqrt[n]{c})^m$ " is true. The proof involves rewriting the expression $c^{m / n}$ as $(c^{1/n})^m$ and recognizing that it is equivalent to $\sqrt[n]{c^m}$ . The example illustrates the statement with specific values of $c$ , $m$ , and $n$ . This statement has important implications in algebra and beyond, and it is a fundamental concept in mathematics.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about exponents and roots.

Q: What is the difference between an exponent and a root?

A: An exponent is a small number that is written above and to the right of a base number. It represents the number of times the base number is multiplied by itself. For example, $2^3$ means $2$ multiplied by itself $3$ times, which equals $8$ . A root, on the other hand, is the inverse operation of an exponent. It represents the number that, when raised to a certain power, equals a given value. For example, $\sqrt{16}$ means the number that, when squared, equals $16$ , which is $4$ .

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can use the following properties:

Product of Powers: When multiplying two numbers with the same base, add the exponents. For example, $2^3 \cdot 2^4 = 2^{3+4} = 2^7$
Power of a Power: When raising a number with an exponent to another power, multiply the exponents. For example, $(2^3)^4 = 2^{3\cdot4} = 2^{12}$
Zero Exponent: Any number raised to the power of zero is equal to $1$ . For example, $2^0 = 1$

Q: How do I simplify an expression with roots?

A: To simplify an expression with roots, you can use the following properties:

Product of Roots: When multiplying two numbers with the same root, multiply the numbers inside the root. For example, $\sqrt{2} \cdot \sqrt{3} = \sqrt{2\cdot3} = \sqrt{6}$
Power of a Root: When raising a number with a root to another power, multiply the exponents. For example, $(\sqrt{2})^4 = 2^{4/2} = 2^2 = 4$
Negative Exponent: Any number raised to a negative power is equal to the reciprocal of the number raised to the positive power. For example, $2^{-3} = 1/2^3 = 1/8$

Q: How do I evaluate an expression with exponents and roots?

A: To evaluate an expression with exponents and roots, you can use the following steps:

Simplify the exponents: Use the properties of exponents to simplify the expression.
Simplify the roots: Use the properties of roots to simplify the expression.
Evaluate the expression: Use the simplified expression to evaluate the original expression.

Q: What are some common mistakes to avoid when working with exponents and roots?

A: Some common mistakes to avoid when working with exponents and roots include:

Forgetting to simplify the exponents: Make sure to simplify the exponents before evaluating the expression.
Forgetting to simplify the roots: Make sure to simplify the roots before evaluating the expression.
Not using the correct properties: Make sure to use the correct properties of exponents and roots to simplify the expression.

Q: How do I apply exponents and roots in real-world problems?

A: Exponents and roots are used in a wide range of real-world problems, including:

Finance: Exponents and roots are used to calculate interest rates and investment returns.
Science: Exponents and roots are used to describe the growth and decay of populations and chemical reactions.
Engineering: Exponents and roots are used to design and optimize systems and structures.

Conclusion

In conclusion, exponents and roots are fundamental concepts in mathematics that have a wide range of applications in real-world problems. By understanding the properties and rules of exponents and roots, you can simplify complex expressions and evaluate them accurately. Remember to avoid common mistakes and apply the correct properties to simplify the expression. With practice and experience, you will become proficient in working with exponents and roots and be able to apply them in a variety of real-world problems.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

If $c$ Is A Positive Real Number And $m$ And $n$ Are Positive Integers, Then $c^{m / N}=\sqrt[n]{c^m}=(\sqrt[n]{c})^m$.A. True B. False

The Basics of Exponents and Roots

Understanding the Statement

Proof of the Statement

Example

Conclusion

References

Further Reading

Frequently Asked Questions

Q: What is the difference between an exponent and a root?

Q: How do I simplify an expression with exponents?

Q: How do I simplify an expression with roots?

Q: How do I evaluate an expression with exponents and roots?

Q: What are some common mistakes to avoid when working with exponents and roots?

Q: How do I apply exponents and roots in real-world problems?

Conclusion

References

Further Reading