Applying the Definition to the Problem

Oct 13, 2025 by miodeya.com 25 views

Understanding the Problem

When dealing with radical expressions, it's essential to understand the properties of exponents and how they interact with roots. In this problem, we're given the expression $(\sqrt[4]{6})^3$ and asked to rewrite it in exponential form. To approach this, we need to recall the definition of a root and how it relates to exponents.

The Definition of a Root

A root of a number is a value that, when raised to a certain power, gives the original number. For example, the square root of 16 is 4, because $4^2 = 16$ . Similarly, the cube root of 27 is 3, because $3^3 = 27$ . In general, the nth root of a number x is a value that, when raised to the power of n, gives x.

Applying the Definition to the Problem

In the given expression, we have the fourth root of 6, which means that there is a value that, when raised to the power of 4, gives 6. This value is denoted by $\sqrt[4]{6}$ . When we raise this value to the power of 3, we are essentially raising the fourth root of 6 to the power of 3.

Using Exponent Rules to Simplify

To rewrite the expression in exponential form, we can use the rule that states $(a^m)^n = a^{mn}$ . In this case, we have $(\sqrt[4]{6})^3$ , which can be rewritten as $(6^{\frac{1}{4}})^3$ . Applying the exponent rule, we get $6^{\frac{3}{4}}$ .

Conclusion

Therefore, the radical expression $(\sqrt[4]{6})^3$ is written in exponential form as $6^{\frac{3}{4}}$ . This is the correct answer among the options provided.

Why the Other Options Are Incorrect

Let's take a look at the other options to see why they are incorrect.

Option B, $6^1$ , is incorrect because it does not take into account the root of 6. The expression $(\sqrt[4]{6})^3$ is not simply equal to 6 raised to the power of 1.
Option C, $6^{\frac{4}{3}}$ , is incorrect because it reverses the order of the exponents. The correct order is to raise the fourth root of 6 to the power of 3, not the other way around.
Option D, $6^{12}$ , is incorrect because it does not take into account the root of 6. The expression $(\sqrt[4]{6})^3$ is not simply equal to 6 raised to the power of 12.

Final Thoughts

In conclusion, the radical expression $(\sqrt[4]{6})^3$ is written in exponential form as $6^{\frac{3}{4}}$ . This is a fundamental concept in algebra and requires a deep understanding of exponents and roots. By applying the exponent rule and using the definition of a root, we can simplify the expression and arrive at the correct answer.

Frequently Asked Questions

Q: What is the definition of a root? A: A root of a number is a value that, when raised to a certain power, gives the original number.
Q: How do I rewrite a radical expression in exponential form? A: To rewrite a radical expression in exponential form, you can use the rule that states $(a^m)^n = a^{mn}$ .
Q: What is the correct answer among the options provided? A: The correct answer is $6^{\frac{3}{4}}$ .

Additional Resources

For more information on exponents and roots, see the Khan Academy video on "Exponents and Roots".
For more practice problems on rewriting radical expressions in exponential form, see the Mathway website.

Conclusion

In this article, we have explored the concept of rewriting radical expressions in exponential form. We have applied the exponent rule and used the definition of a root to simplify the expression $(\sqrt[4]{6})^3$ and arrive at the correct answer, $6^{\frac{3}{4}}$ . This is a fundamental concept in algebra and requires a deep understanding of exponents and roots. By following the steps outlined in this article, you can master this concept and become proficient in rewriting radical expressions in exponential form.

Understanding Radical Expressions and Exponential Form

Radical expressions and exponential form are fundamental concepts in algebra that require a deep understanding of exponents and roots. In this article, we will explore some of the most frequently asked questions about radical expressions and exponential form.

Q: What is the difference between a radical expression and an exponential expression?

A: A radical expression is an expression that contains a root, such as $\sqrt{x}$ or $\sqrt[3]{x}$ . An exponential expression, on the other hand, is an expression that contains an exponent, such as $x^2$ or $x^3$ .

Q: How do I rewrite a radical expression in exponential form?

A: To rewrite a radical expression in exponential form, you can use the rule that states $(a^m)^n = a^{mn}$ . For example, to rewrite $\sqrt[4]{x}$ in exponential form, you can raise it to the power of 4, resulting in $x^{\frac{1}{4}}$ .

Q: What is the relationship between radical expressions and exponential form?

A: Radical expressions and exponential form are closely related. In fact, they are two sides of the same coin. A radical expression can be rewritten in exponential form, and vice versa. For example, the radical expression $\sqrt{x}$ can be rewritten in exponential form as $x^{\frac{1}{2}}$ .

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can use the following steps:

Identify the root of the expression.
Raise the root to the power of the exponent.
Simplify the resulting expression.

For example, to simplify $\sqrt[3]{x^2}$ , you can raise the root to the power of 3, resulting in $x^2$ .

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

A: A rational exponent is an exponent that can be expressed as a fraction, such as $\frac{1}{2}$ or $\frac{3}{4}$ . An irrational exponent, on the other hand, is an exponent that cannot be expressed as a fraction, such as $\sqrt{2}$ or $\pi$ .

Q: How do I handle rational exponents?

A: To handle rational exponents, you can use the following steps:

Identify the numerator and denominator of the exponent.
Raise the base to the power of the numerator.
Take the root of the result to the power of the denominator.

For example, to handle the rational exponent $\frac{1}{2}$ , you can raise the base to the power of 1 and take the square root of the result.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent is an exponent that is greater than 0, such as 2 or 3. A negative exponent, on the other hand, is an exponent that is less than 0, such as -2 or -3.

Q: How do I handle negative exponents?

A: To handle negative exponents, you can use the following steps:

Identify the negative exponent.
Take the reciprocal of the base.
Raise the reciprocal to the power of the absolute value of the exponent.

For example, to handle the negative exponent -2, you can take the reciprocal of the base and raise it to the power of 2.

Q: What is the relationship between radical expressions and negative exponents?

A: Radical expressions and negative exponents are closely related. In fact, they are two sides of the same coin. A radical expression can be rewritten in terms of a negative exponent, and vice versa. For example, the radical expression $\sqrt{x}$ can be rewritten in terms of a negative exponent as $x^{-\frac{1}{2}}$ .

Q: How do I simplify a radical expression with a negative exponent?

A: To simplify a radical expression with a negative exponent, you can use the following steps:

Identify the negative exponent.
Take the reciprocal of the base.
Raise the reciprocal to the power of the absolute value of the exponent.
Simplify the resulting expression.

For example, to simplify $\sqrt{x^{-2}}$ , you can take the reciprocal of the base, raise it to the power of 2, and simplify the resulting expression.

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

A: A rational root is a root that can be expressed as a fraction, such as $\frac{1}{2}$ or $\frac{3}{4}$ . An irrational root, on the other hand, is a root that cannot be expressed as a fraction, such as $\sqrt{2}$ or $\pi$ .

Q: How do I handle rational roots?

A: To handle rational roots, you can use the following steps:

Identify the numerator and denominator of the root.
Raise the base to the power of the numerator.
Take the root of the result to the power of the denominator.

For example, to handle the rational root $\frac{1}{2}$ , you can raise the base to the power of 1 and take the square root of the result.

Q: What is the relationship between radical expressions and rational roots?

A: Radical expressions and rational roots are closely related. In fact, they are two sides of the same coin. A radical expression can be rewritten in terms of a rational root, and vice versa. For example, the radical expression $\sqrt{x}$ can be rewritten in terms of a rational root as $x^{\frac{1}{2}}$ .

Q: How do I simplify a radical expression with a rational root?

A: To simplify a radical expression with a rational root, you can use the following steps:

Identify the rational root.
Raise the base to the power of the numerator.
Take the root of the result to the power of the denominator.
Simplify the resulting expression.

For example, to simplify $\sqrt{x^{\frac{1}{2}}}$ , you can raise the base to the power of 1 and take the square root of the result.

Conclusion

In this article, we have explored some of the most frequently asked questions about radical expressions and exponential form. We have discussed the relationship between radical expressions and exponential form, and how to handle rational exponents, negative exponents, and rational roots. By following the steps outlined in this article, you can master the concepts of radical expressions and exponential form and become proficient in simplifying complex expressions.