Rewriting Radical Expressions in Rational Exponent Form

Oct 13, 2025 by miodeya.com 124 views

Understanding Radical Expressions

Radical expressions are mathematical expressions that contain a root or a radical sign. They are often used to represent the square root or other roots of a number. In this article, we will focus on rewriting a radical expression in rational exponent form.

Rewriting Radical Expressions in Rational Exponent Form

To rewrite a radical expression in rational exponent form, we need to understand the properties of exponents and radicals. The property we will use is:

\sqrt[n]{x} = x^{\frac{1}{n}}

This property states that the nth root of x is equal to x raised to the power of 1/n.

Rewriting the Given Radical Expression

The given radical expression is $(\sqrt{x})^5$ . To rewrite this expression in rational exponent form, we can use the property mentioned above.

\sqrt{x} = x^{\frac{1}{2}}

Now, we can substitute this expression into the original expression:

(\sqrt{x})^5 = (x^{\frac{1}{2}})^5

Using the property of exponents, we can rewrite this expression as:

x^{\frac{1}{2} \cdot 5} = x^{\frac{5}{2}}

Therefore, the correct answer is B. $x^{\frac{5}{2}}$ .

Explanation of the Other Options

Let's take a look at the other options and explain why they are incorrect.

Option A: $\frac{x^2}{x^5}$

This option is incorrect because it does not follow the property of exponents and radicals. The correct way to rewrite the expression is to use the property mentioned above.

Option C: ${ }_x{ }^{\frac{2}{5}}$

This option is incorrect because it is not in the correct format. The correct format for a rational exponent is $x^{\frac{a}{b}}$ , where a and b are integers.

Option D: $\left(\frac{1}{x^2}\right)^5$

This option is incorrect because it does not follow the property of exponents and radicals. The correct way to rewrite the expression is to use the property mentioned above.

Conclusion

In conclusion, the correct answer is B. $x^{\frac{5}{2}}$ . This is because it follows the property of exponents and radicals, and it is in the correct format.

Common Mistakes to Avoid

When rewriting radical expressions in rational exponent form, there are several common mistakes to avoid.

Not following the property of exponents and radicals: This is the most common mistake. Make sure to use the property mentioned above to rewrite the expression.
Not using the correct format: Make sure to use the correct format for a rational exponent, which is $x^{\frac{a}{b}}$ , where a and b are integers.
Not simplifying the expression: Make sure to simplify the expression after rewriting it in rational exponent form.

Practice Problems

Here are some practice problems to help you understand how to rewrite radical expressions in rational exponent form.

Rewrite the radical expression $\sqrt[3]{x}$ in rational exponent form.
Rewrite the radical expression $(\sqrt{x})^4$ in rational exponent form.
Rewrite the radical expression $\sqrt[5]{x^3}$ in rational exponent form.

Answer Key

$x^{\frac{1}{3}}$
$x^2$
$x^{\frac{3}{5}}$

Conclusion

Frequently Asked Questions

In this article, we will answer some frequently asked questions about rewriting radical expressions in rational exponent form.

Q: What is the property of exponents and radicals that we need to use to rewrite radical expressions in rational exponent form?

A: The property of exponents and radicals that we need to use is:

\sqrt[n]{x} = x^{\frac{1}{n}}

This property states that the nth root of x is equal to x raised to the power of 1/n.

Q: How do we rewrite a radical expression in rational exponent form?

A: To rewrite a radical expression in rational exponent form, we need to follow these steps:

Identify the root of the radical expression.
Use the property of exponents and radicals to rewrite the radical expression in rational exponent form.
Simplify the expression.

Q: What is the correct format for a rational exponent?

A: The correct format for a rational exponent is $x^{\frac{a}{b}}$ , where a and b are integers.

Q: How do we simplify a rational exponent?

A: To simplify a rational exponent, we need to follow these steps:

Identify the numerator and denominator of the rational exponent.
Simplify the numerator and denominator separately.
Divide the numerator by the denominator.

Q: What are some common mistakes to avoid when rewriting radical expressions in rational exponent form?

A: Some common mistakes to avoid when rewriting radical expressions in rational exponent form include:

Not following the property of exponents and radicals.
Not using the correct format for a rational exponent.
Not simplifying the expression.

Q: How do we rewrite a radical expression with a variable in the radicand in rational exponent form?

A: To rewrite a radical expression with a variable in the radicand in rational exponent form, we need to follow these steps:

Identify the variable in the radicand.
Use the property of exponents and radicals to rewrite the radical expression in rational exponent form.
Simplify the expression.

Q: How do we rewrite a radical expression with a coefficient in the radicand in rational exponent form?

A: To rewrite a radical expression with a coefficient in the radicand in rational exponent form, we need to follow these steps:

Identify the coefficient in the radicand.
Use the property of exponents and radicals to rewrite the radical expression in rational exponent form.
Simplify the expression.

Q: What are some examples of radical expressions that can be rewritten in rational exponent form?

A: Some examples of radical expressions that can be rewritten in rational exponent form include:

$\sqrt{x}$
$(\sqrt{x})^4$
$\sqrt[5]{x^3}$

Q: How do we check our work when rewriting radical expressions in rational exponent form?

A: To check your work when rewriting radical expressions in rational exponent form, you can follow these steps:

Rewrite the radical expression in rational exponent form.
Simplify the expression.
Check that the simplified expression is equivalent to the original radical expression.

Conclusion

In conclusion, rewriting radical expressions in rational exponent form is an important skill in mathematics. By following the property of exponents and radicals, and using the correct format, you can rewrite radical expressions in rational exponent form. Practice problems are also included to help you understand how to rewrite radical expressions in rational exponent form.