Select The Equivalent Expression. A − 6 A 2 16 \sqrt[16]{\frac{a^{-6}}{a^2}} 16 A 2 A − 6 ​ ​ Answer:A. A − 2 A^{-2} A − 2 B. A 1 2 A^{\frac{1}{2}} A 2 1 ​ C. A 2 A^2 A 2 D. A − 1 2 A^{-\frac{1}{2}} A − 2 1 ​

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students to master. In this article, we will focus on simplifying a specific radical expression, a−6a216\sqrt[16]{\frac{a^{-6}}{a^2}}, and explore the different options available to us.

Understanding the Problem

The given expression is a−6a216\sqrt[16]{\frac{a^{-6}}{a^2}}. To simplify this expression, we need to apply the rules of exponents and radicals. The expression involves a fraction with negative and positive exponents, as well as a radical with a fractional index.

Applying the Rules of Exponents

To simplify the expression, we need to start by applying the rules of exponents. We can use the rule aman=am−n\frac{a^m}{a^n} = a^{m-n} to simplify the fraction inside the radical.

a−6a2=a−6−2=a−8\frac{a^{-6}}{a^2} = a^{-6-2} = a^{-8}

Now, we can rewrite the original expression as:

a−816\sqrt[16]{a^{-8}}

Applying the Rules of Radicals

Next, we need to apply the rules of radicals. We can use the rule amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}} to simplify the radical expression.

a−816=a−816=a−12\sqrt[16]{a^{-8}} = a^{\frac{-8}{16}} = a^{-\frac{1}{2}}

Evaluating the Options

Now that we have simplified the expression, we can evaluate the options available to us.

  • Option A: a−2a^{-2}
  • Option B: a12a^{\frac{1}{2}}
  • Option C: a2a^2
  • Option D: a−12a^{-\frac{1}{2}}

Based on our simplification, we can see that the correct answer is Option D: a−12a^{-\frac{1}{2}}.

Conclusion

In this article, we simplified a radical expression using the rules of exponents and radicals. We started by applying the rules of exponents to simplify the fraction inside the radical, and then applied the rules of radicals to simplify the radical expression. By following these steps, we were able to evaluate the options available to us and determine the correct answer.

Key Takeaways

  • To simplify a radical expression, we need to apply the rules of exponents and radicals.
  • We can use the rule aman=am−n\frac{a^m}{a^n} = a^{m-n} to simplify the fraction inside the radical.
  • We can use the rule amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}} to simplify the radical expression.
  • By following these steps, we can simplify complex radical expressions and evaluate the options available to us.

Frequently Asked Questions

  • What is the rule for simplifying a fraction inside a radical?
    • The rule is aman=am−n\frac{a^m}{a^n} = a^{m-n}.
  • What is the rule for simplifying a radical expression?
    • The rule is amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}.
  • How do I simplify a complex radical expression?
    • Start by applying the rules of exponents to simplify the fraction inside the radical, and then apply the rules of radicals to simplify the radical expression.

Additional Resources

  • For more information on simplifying radical expressions, check out the following resources:
    • Khan Academy: Simplifying Radical Expressions
    • Mathway: Simplifying Radical Expressions
    • Purplemath: Simplifying Radical Expressions
      Simplifying Radical Expressions: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the concept of simplifying radical expressions using the rules of exponents and radicals. In this article, we will delve deeper into the topic and provide a Q&A guide to help you better understand the concept.

Q&A: Simplifying Radical Expressions

Q: What is the rule for simplifying a fraction inside a radical?

A: The rule is aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Q: What is the rule for simplifying a radical expression?

A: The rule is amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}.

Q: How do I simplify a complex radical expression?

A: Start by applying the rules of exponents to simplify the fraction inside the radical, and then apply the rules of radicals to simplify the radical expression.

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

A: A radical is a symbol that represents the square root of a number, while an exponent is a number that represents the power to which a number is raised.

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 rule a−mn=a−mn\sqrt[n]{a^{-m}} = a^{-\frac{m}{n}}.

Q: Can I simplify a radical expression with a fractional exponent?

A: Yes, you can simplify a radical expression with a fractional exponent using the rule ampn=amnp\sqrt[n]{a^{\frac{m}{p}}} = a^{\frac{m}{np}}.

Q: What is the order of operations for simplifying radical expressions?

A: The order of operations is:

  1. Simplify the fraction inside the radical using the rule aman=am−n\frac{a^m}{a^n} = a^{m-n}.
  2. Simplify the radical expression using the rule amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}.
  3. Simplify any remaining exponents or radicals.

Q: Can I use a calculator to simplify radical expressions?

A: Yes, you can use a calculator to simplify radical expressions. However, it's always a good idea to double-check your work by simplifying the expression manually.

Q: What are some common mistakes to avoid when simplifying radical expressions?

A: Some common mistakes to avoid when simplifying radical expressions include:

  • Forgetting to simplify the fraction inside the radical.
  • Forgetting to simplify the radical expression.
  • Not following the order of operations.
  • Not checking your work.

Conclusion

Simplifying radical expressions is an important concept in mathematics, and it requires a good understanding of the rules of exponents and radicals. By following the steps outlined in this article and practicing regularly, you can become proficient in simplifying radical expressions and tackle even the most complex problems.

Key Takeaways

  • To simplify a radical expression, you need to apply the rules of exponents and radicals.
  • The rule for simplifying a fraction inside a radical is aman=am−n\frac{a^m}{a^n} = a^{m-n}.
  • The rule for simplifying a radical expression is amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}.
  • The order of operations for simplifying radical expressions is to simplify the fraction inside the radical, then simplify the radical expression, and finally simplify any remaining exponents or radicals.

Frequently Asked Questions

  • What is the rule for simplifying a fraction inside a radical?
    • The rule is aman=am−n\frac{a^m}{a^n} = a^{m-n}.
  • What is the rule for simplifying a radical expression?
    • The rule is amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}.
  • How do I simplify a complex radical expression?
    • Start by applying the rules of exponents to simplify the fraction inside the radical, and then apply the rules of radicals to simplify the radical expression.

Additional Resources

  • For more information on simplifying radical expressions, check out the following resources:
    • Khan Academy: Simplifying Radical Expressions
    • Mathway: Simplifying Radical Expressions
    • Purplemath: Simplifying Radical Expressions