Use The Properties Of Rational Exponents To Simplify The Expression. ( 12 4 ) 1 5 \left(12^4\right)^{\frac{1}{5}} ( 1 2 4 ) 5 1 A. 12 4 5 12^{\frac{4}{5}} 1 2 5 4 B. 12 20 12^{20} 1 2 20 C. 12 21 5 12^{\frac{21}{5}} 1 2 5 21 D. 1 12 20 \frac{1}{12^{20}} 1 2 20 1

Mar 11, 2025 by ADMIN 271 views

**Simplifying Rational Exponents: A Step-by-Step Guide**

Introduction

Rational exponents are a powerful tool in algebra, allowing us to simplify complex expressions and solve equations with ease. In this article, we will explore the properties of rational exponents and use them to simplify the expression $\left(12^4\right)^{\frac{1}{5}}$ . We will examine the different options available and determine the correct solution.

Understanding Rational Exponents

Rational exponents are a way of expressing a number raised to a power that is itself a fraction. The general form of a rational exponent is $a^{\frac{m}{n}}$ , where $a$ is the base, $m$ is the numerator, and $n$ is the denominator. Rational exponents can be used to simplify expressions and solve equations.

Properties of Rational Exponents

There are several properties of rational exponents that we need to understand in order to simplify the expression $\left(12^4\right)^{\frac{1}{5}}$ . These properties include:

Product of Powers: When we multiply two numbers with rational exponents, we can add their exponents. For example, $\left(a^m\right)\left(a^n\right) = a^{m+n}$ .
Power of a Power: When we raise a number with a rational exponent to another power, we can multiply the exponents. For example, $\left(a^m\right)^n = a^{mn}$ .
Zero Exponent: Any number raised to the power of zero is equal to 1. For example, $a^0 = 1$ .

Simplifying the Expression

Now that we have a good understanding of rational exponents and their properties, we can simplify the expression $\left(12^4\right)^{\frac{1}{5}}$ . Using the property of power of a power, we can rewrite the expression as:

\left(12^4\right)^{\frac{1}{5}} = 12^{4 \cdot \frac{1}{5}}

Simplifying the exponent, we get:

12^{4 \cdot \frac{1}{5}} = 12^{\frac{4}{5}}

Therefore, the simplified expression is $12^{\frac{4}{5}}$ .

Checking the Options

Now that we have simplified the expression, we can check the options available to see which one matches our solution.

Option A: $12^{\frac{4}{5}}$ - This option matches our solution.
Option B: $12^{20}$ - This option does not match our solution.
Option C: $12^{\frac{21}{5}}$ - This option does not match our solution.
Option D: $\frac{1}{12^{20}}$ - This option does not match our solution.

Conclusion

In this article, we used the properties of rational exponents to simplify the expression $\left(12^4\right)^{\frac{1}{5}}$ . We examined the different options available and determined that the correct solution is $12^{\frac{4}{5}}$ . We also discussed the properties of rational exponents and how they can be used to simplify complex expressions and solve equations.

Final Answer

Q: What is a rational exponent?

A: A rational exponent is a way of expressing a number raised to a power that is itself a fraction. The general form of a rational exponent is $a^{\frac{m}{n}}$ , where $a$ is the base, $m$ is the numerator, and $n$ is the denominator.

Q: How do I simplify a rational exponent?

A: To simplify a rational exponent, you can use the properties of rational exponents, such as the product of powers, power of a power, and zero exponent. For example, if you have the expression $\left(a^m\right)^n$ , you can simplify it by multiplying the exponents: $\left(a^m\right)^n = a^{mn}$ .

Q: What is the product of powers property?

A: The product of powers property states that when you multiply two numbers with rational exponents, you can add their exponents. For example, $\left(a^m\right)\left(a^n\right) = a^{m+n}$ .

Q: What is the power of a power property?

A: The power of a power property states that when you raise a number with a rational exponent to another power, you can multiply the exponents. For example, $\left(a^m\right)^n = a^{mn}$ .

Q: What is the zero exponent property?

A: The zero exponent property states that any number raised to the power of zero is equal to 1. For example, $a^0 = 1$ .

Q: How do I apply the properties of rational exponents to simplify an expression?

A: To apply the properties of rational exponents, you need to identify the base, numerator, and denominator of the rational exponent. Then, you can use the properties to simplify the expression. For example, if you have the expression $\left(a^m\right)^n$ , you can simplify it by multiplying the exponents: $\left(a^m\right)^n = a^{mn}$ .

Q: What are some common mistakes to avoid when simplifying rational exponents?

A: Some common mistakes to avoid when simplifying rational exponents include:

Not identifying the base, numerator, and denominator of the rational exponent
Not applying the properties of rational exponents correctly
Not simplifying the expression fully

Q: How do I check my answer when simplifying a rational exponent?

A: To check your answer when simplifying a rational exponent, you can plug in a value for the base and see if the expression simplifies to the correct value. For example, if you have the expression $\left(a^m\right)^n$ , you can plug in a value for $a$ and see if the expression simplifies to $a^{mn}$ .

Q: What are some real-world applications of rational exponents?

A: Rational exponents have many real-world applications, including:

Physics: Rational exponents are used to describe the motion of objects and the behavior of physical systems.
Engineering: Rational exponents are used to design and optimize systems, such as electrical circuits and mechanical systems.
Computer Science: Rational exponents are used in algorithms and data structures to solve problems and optimize performance.

Conclusion

In this article, we answered some frequently asked questions about rational exponents, including how to simplify them, how to apply the properties of rational exponents, and how to check your answer. We also discussed some common mistakes to avoid and some real-world applications of rational exponents.