Which Of The Following Is Equivalent To $(\sqrt[4]{3})^7$?A. $4^{3 / 7}$B. $ 3 7 / 4 3^{7 / 4} 3 7/4 [/tex]C. $3^{4 / 7}$D. $4^{7 / 3}$

Mar 9, 2025 by ADMIN 144 views

**Simplifying Radical Expressions: A Step-by-Step Guide**

Understanding the Problem

In this article, we will explore the concept of simplifying radical expressions and apply it to a specific problem. The problem asks us to find the equivalent expression for $(\sqrt[4]{3})^7$. To solve this problem, we need to understand the properties of exponents and radicals.

Properties of Exponents and Radicals

Before we dive into the problem, let's review some key properties of exponents and radicals.

Power of a Power Rule: When we raise a power to a power, we multiply the exponents. For example, $(a^m)n = a^{mn}$.
Product of Powers Rule: When we multiply powers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
Quotient of Powers Rule: When we divide powers with the same base, we subtract the exponents. For example, $\frac{a^m}{an} = a^{m-n}$.
Roots and Exponents: A radical expression can be rewritten as an exponential expression. For example, $\sqrt[n]{a} = a^{1/n}$.

Simplifying Radical Expressions

Now that we have reviewed the properties of exponents and radicals, let's apply them to the problem.

We are given the expression $(\sqrt[4]{3})^7$. To simplify this expression, we can use the power of a power rule.

(\sqrt[4]{3})^7 = (3^{1/4})^7

Using the power of a power rule, we can rewrite this expression as:

(3^{1/4})^7 = 3^{7/4}

Comparing the Options

Now that we have simplified the expression, let's compare it to the options.

A. $4^{3 / 7}$ B. $3^{7 / 4}$ C. $3^{4 / 7}$ D. $4^{7 / 3}$

Based on our simplification, we can see that option B is the correct answer.

Conclusion

In this article, we have explored the concept of simplifying radical expressions and applied it to a specific problem. We have reviewed the properties of exponents and radicals and used them to simplify the expression $(\sqrt[4]{3})^7$. We have also compared the simplified expression to the options and found that option B is the correct answer.

Final Answer

The final answer is:

B. $3^{7 / 4}$

Additional Resources

If you want to learn more about simplifying radical expressions, here are some additional resources:

Khan Academy: Simplifying Radical Expressions
Mathway: Simplifying Radical Expressions
Wolfram Alpha: Simplifying Radical Expressions

Frequently Asked Questions

Here are some frequently asked questions about simplifying radical expressions:

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 $\sqrt[n]{a}$. An exponential expression is an expression that contains an exponent, such as $a^m$.
Q: How do I simplify a radical expression?
A: To simplify a radical expression, you can use the properties of exponents and radicals, such as the power of a power rule, the product of powers rule, and the quotient of powers rule.
Q: What is the power of a power rule?
A: The power of a power rule states that when we raise a power to a power, we multiply the exponents. For example, $(a^m)n = a^{mn}$.

Glossary

Here are some key terms related to simplifying radical expressions:

Radical expression: An expression that contains a root, such as $\sqrt[n]{a}$.
Exponential expression: An expression that contains an exponent, such as $a^m$.
Power of a power rule: A rule that states that when we raise a power to a power, we multiply the exponents. For example, $(a^m)n = a^{mn}$.
Product of powers rule: A rule that states that when we multiply powers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
Quotient of powers rule: A rule that states that when we divide powers with the same base, we subtract the exponents. For example, $\frac{a^m}{an} = a^{m-n}$.

References

Here are some references related to simplifying radical expressions:

Khan Academy: Simplifying Radical Expressions
Mathway: Simplifying Radical Expressions
Wolfram Alpha: Simplifying Radical Expressions

About the Author

Frequently Asked Questions

In this article, we will answer some frequently asked questions about simplifying radical expressions.

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 $\sqrt[n]{a}$. An exponential expression is an expression that contains an exponent, such as $a^m$.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can use the properties of exponents and radicals, such as the power of a power rule, the product of powers rule, and the quotient of powers rule.

Q: What is the power of a power rule?

A: The power of a power rule states that when we raise a power to a power, we multiply the exponents. For example, $(a^m)n = a^{mn}$.

Q: What is the product of powers rule?

A: The product of powers rule states that when we multiply powers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: What is the quotient of powers rule?

A: The quotient of powers rule states that when we divide powers with the same base, we subtract the exponents. For example, $\frac{a^m}{an} = a^{m-n}$.

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 property of negative exponents, which states that $a^{-m} = \frac{1}{a^m}$.

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

A: To simplify a radical expression with a fractional exponent, you can use the property of fractional exponents, which states that $a^{m/n} = \sqrt[n]{a^m}$.

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 $\frac{m}{n}$. An irrational exponent is an exponent that cannot be expressed as a fraction, such as $\sqrt{2}$.

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

A: To simplify a radical expression with a rational exponent, you can use the property of rational exponents, which states that $a^{m/n} = \sqrt[n]{a^m}$.

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

A: To simplify a radical expression with an irrational exponent, you cannot simplify it further using the properties of exponents and radicals.

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

A: A radical expression is an expression that contains a root, such as $\sqrt[n]{a}$. A rational expression is an expression that contains a fraction, such as $\frac{a}{b}$.

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

A: To simplify a radical expression with a rational expression, you can use the properties of exponents and radicals, such as the power of a power rule, the product of powers rule, and the quotient of powers rule.

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

A: A radical expression is an expression that contains a root, such as $\sqrt[n]{a}$. An algebraic expression is an expression that contains variables and constants, such as $2x + 3$.

Q: How do I simplify a radical expression with an algebraic expression?

A: To simplify a radical expression with an algebraic expression, you can use the properties of exponents and radicals, such as the power of a power rule, the product of powers rule, and the quotient of powers rule.

Conclusion

In this article, we have answered some frequently asked questions about simplifying radical expressions. We have covered topics such as the difference between a radical expression and an exponential expression, the power of a power rule, the product of powers rule, and the quotient of powers rule. We have also covered topics such as simplifying radical expressions with negative exponents, fractional exponents, rational exponents, and irrational exponents. We have also covered topics such as simplifying radical expressions with rational expressions and algebraic expressions.

Final Answer

The final answer is:

There is no final answer, as this is a Q&A guide.

Additional Resources

If you want to learn more about simplifying radical expressions, here are some additional resources:

Khan Academy: Simplifying Radical Expressions
Mathway: Simplifying Radical Expressions
Wolfram Alpha: Simplifying Radical Expressions

Glossary

Here are some key terms related to simplifying radical expressions:

Radical expression: An expression that contains a root, such as $\sqrt[n]{a}$.
Exponential expression: An expression that contains an exponent, such as $a^m$.
Power of a power rule: A rule that states that when we raise a power to a power, we multiply the exponents. For example, $(a^m)n = a^{mn}$.
Product of powers rule: A rule that states that when we multiply powers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
Quotient of powers rule: A rule that states that when we divide powers with the same base, we subtract the exponents. For example, $\frac{a^m}{an} = a^{m-n}$.

References

Here are some references related to simplifying radical expressions:

Khan Academy: Simplifying Radical Expressions
Mathway: Simplifying Radical Expressions
Wolfram Alpha: Simplifying Radical Expressions

About the Author

The author of this article is a math enthusiast who loves to share knowledge about math concepts, including simplifying radical expressions. If you have any questions or comments, feel free to reach out!