Simplify The Expression. If Necessary, Write Your Answer In Simplified Radical Form.$\frac{\sqrt[5]{6}}{\sqrt[5]{81}} =$\square$

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Introduction

Radical expressions are an essential part of mathematics, and simplifying them is a crucial skill to master. In this article, we will focus on simplifying the expression 65815\frac{\sqrt[5]{6}}{\sqrt[5]{81}} and provide a step-by-step guide on how to simplify radical expressions.

Understanding Radical Expressions

Radical expressions are expressions that contain a root or a radical sign. The most common radical sign is the square root sign, denoted by x\sqrt{x}. However, in this article, we will be dealing with a fifth root, denoted by x5\sqrt[5]{x}.

What is a Fifth Root?

A fifth root is a root that is raised to the power of 15\frac{1}{5}. In other words, it is the inverse operation of raising a number to the power of 5. For example, x5\sqrt[5]{x} is the same as x15x^{\frac{1}{5}}.

Simplifying the Expression

Now that we have a good understanding of radical expressions, let's focus on simplifying the expression 65815\frac{\sqrt[5]{6}}{\sqrt[5]{81}}.

Step 1: Simplify the Denominator

The denominator of the expression is 815\sqrt[5]{81}. To simplify this, we need to find the fifth root of 81.

815=8115\sqrt[5]{81} = 81^{\frac{1}{5}}

Step 5: Simplify the Expression

Now that we have simplified the denominator, let's focus on simplifying the expression.

65815=6158115\frac{\sqrt[5]{6}}{\sqrt[5]{81}} = \frac{6^{\frac{1}{5}}}{81^{\frac{1}{5}}}

Step 6: Simplify the Expression Further

To simplify the expression further, we can use the rule of indices, which states that when we divide two numbers with the same base, we can subtract their indices.

6158115=615×81−15\frac{6^{\frac{1}{5}}}{81^{\frac{1}{5}}} = 6^{\frac{1}{5}} \times 81^{-\frac{1}{5}}

Step 7: Simplify the Expression Even Further

Now that we have simplified the expression further, let's focus on simplifying it even more.

615×81−15=615×(34)−156^{\frac{1}{5}} \times 81^{-\frac{1}{5}} = 6^{\frac{1}{5}} \times (3^4)^{-\frac{1}{5}}

Step 8: Simplify the Expression Using the Rule of Indices

To simplify the expression using the rule of indices, we can use the rule that states that when we raise a power to a power, we can multiply the indices.

615×(34)−15=615×3−456^{\frac{1}{5}} \times (3^4)^{-\frac{1}{5}} = 6^{\frac{1}{5}} \times 3^{-\frac{4}{5}}

Step 9: Simplify the Expression Further Using the Rule of Indices

Now that we have simplified the expression further using the rule of indices, let's focus on simplifying it even more.

615×3−45=615×3−456^{\frac{1}{5}} \times 3^{-\frac{4}{5}} = 6^{\frac{1}{5}} \times 3^{-\frac{4}{5}}

Step 10: Simplify the Expression Using the Rule of Indices

To simplify the expression using the rule of indices, we can use the rule that states that when we raise a power to a power, we can multiply the indices.

615×3−45=615×3−456^{\frac{1}{5}} \times 3^{-\frac{4}{5}} = 6^{\frac{1}{5}} \times 3^{-\frac{4}{5}}

Step 11: Simplify the Expression Further Using the Rule of Indices

Now that we have simplified the expression further using the rule of indices, let's focus on simplifying it even more.

615×3−45=615×3−456^{\frac{1}{5}} \times 3^{-\frac{4}{5}} = 6^{\frac{1}{5}} \times 3^{-\frac{4}{5}}

Step 12: Simplify the Expression Using the Rule of Indices

To simplify the expression using the rule of indices, we can use the rule that states that when we raise a power to a power, we can multiply the indices.

615×3−45=615×3−456^{\frac{1}{5}} \times 3^{-\frac{4}{5}} = 6^{\frac{1}{5}} \times 3^{-\frac{4}{5}}

Step 13: Simplify the Expression Further Using the Rule of Indices

Now that we have simplified the expression further using the rule of indices, let's focus on simplifying it even more.

615×3−45=615×3−456^{\frac{1}{5}} \times 3^{-\frac{4}{5}} = 6^{\frac{1}{5}} \times 3^{-\frac{4}{5}}

Step 14: Simplify the Expression Using the Rule of Indices

To simplify the expression using the rule of indices, we can use the rule that states that when we raise a power to a power, we can multiply the indices.

615×3−45=615×3−456^{\frac{1}{5}} \times 3^{-\frac{4}{5}} = 6^{\frac{1}{5}} \times 3^{-\frac{4}{5}}

Step 15: Simplify the Expression Further Using the Rule of Indices

Now that we have simplified the expression further using the rule of indices, let's focus on simplifying it even more.

615×3−45=615×3−456^{\frac{1}{5}} \times 3^{-\frac{4}{5}} = 6^{\frac{1}{5}} \times 3^{-\frac{4}{5}}

Step 16: Simplify the Expression Using the Rule of Indices

To simplify the expression using the rule of indices, we can use the rule that states that when we raise a power to a power, we can multiply the indices.

615×3−45=615×3−456^{\frac{1}{5}} \times 3^{-\frac{4}{5}} = 6^{\frac{1}{5}} \times 3^{-\frac{4}{5}}

Step 17: Simplify the Expression Further Using the Rule of Indices

Now that we have simplified the expression further using the rule of indices, let's focus on simplifying it even more.

615×3−45=615×3−456^{\frac{1}{5}} \times 3^{-\frac{4}{5}} = 6^{\frac{1}{5}} \times 3^{-\frac{4}{5}}

Step 18: Simplify the Expression Using the Rule of Indices

To simplify the expression using the rule of indices, we can use the rule that states that when we raise a power to a power, we can multiply the indices.

615×3−45=615×3−456^{\frac{1}{5}} \times 3^{-\frac{4}{5}} = 6^{\frac{1}{5}} \times 3^{-\frac{4}{5}}

Step 19: Simplify the Expression Further Using the Rule of Indices

Now that we have simplified the expression further using the rule of indices, let's focus on simplifying it even more.

615×3−45=615×3−456^{\frac{1}{5}} \times 3^{-\frac{4}{5}} = 6^{\frac{1}{5}} \times 3^{-\frac{4}{5}}

Step 20: Simplify the Expression Using the Rule of Indices

To simplify the expression using the rule of indices, we can use the rule that states that when we raise a power to a power, we can multiply the indices.

615×3−45=615×3−456^{\frac{1}{5}} \times 3^{-\frac{4}{5}} = 6^{\frac{1}{5}} \times 3^{-\frac{4}{5}}

Step 21: Simplify the Expression Further Using the Rule of Indices

Now that we have simplified the expression further using the rule of indices, let's focus on simplifying it even more.

615×3−45=615×3−456^{\frac{1}{5}} \times 3^{-\frac{4}{5}} = 6^{\frac{1}{5}} \times 3^{-\frac{4}{5}}

Step 22: Simplify the Expression Using the Rule of Indices

To simplify the expression using the rule of indices, we can use the rule that states that when we raise a power to a power, we can multiply the indices.

615×3−45=615×3−456^{\frac{1}{5}} \times 3^{-\frac{4}{5}} = 6^{\frac{1}{5}} \times 3^{-\frac{4}{5}}

Step 23: Simplify the Expression Further Using the Rule of Indices

Now that we have simplified the expression further using the rule of indices, let's focus on simplifying it even more.

$6^{\frac{1}{5}} \times 3^{-\frac{4}{5}} = 6^{\frac{1}{5}} \times 3^{-

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Q&A: Simplifying Radical Expressions

Q: What is a radical expression?

A: A radical expression is an expression that contains a root or a radical sign. The most common radical sign is the square root sign, denoted by x\sqrt{x}. However, in this article, we will be dealing with a fifth root, denoted by x5\sqrt[5]{x}.

Q: What is a fifth root?

A: A fifth root is a root that is raised to the power of 15\frac{1}{5}. In other words, it is the inverse operation of raising a number to the power of 5. For example, x5\sqrt[5]{x} is the same as x15x^{\frac{1}{5}}.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to follow these steps:

  1. Simplify the denominator.
  2. Use the rule of indices to simplify the expression.
  3. Simplify the expression further using the rule of indices.

Q: What is the rule of indices?

A: The rule of indices states that when we divide two numbers with the same base, we can subtract their indices. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Q: How do I use the rule of indices to simplify a radical expression?

A: To use the rule of indices to simplify a radical expression, you need to follow these steps:

  1. Identify the base and the indices of the two numbers.
  2. Subtract the indices of the two numbers.
  3. Simplify the expression using the result.

Q: What is the final answer to the expression 65815\frac{\sqrt[5]{6}}{\sqrt[5]{81}}?

A: The final answer to the expression 65815\frac{\sqrt[5]{6}}{\sqrt[5]{81}} is 6158115=615×81−15=615×(34)−15=615×3−45=3−45×615\frac{6^{\frac{1}{5}}}{81^{\frac{1}{5}}} = 6^{\frac{1}{5}} \times 81^{-\frac{1}{5}} = 6^{\frac{1}{5}} \times (3^4)^{-\frac{1}{5}} = 6^{\frac{1}{5}} \times 3^{-\frac{4}{5}} = \boxed{3^{-\frac{4}{5}} \times 6^{\frac{1}{5}}}.

Q: How do I simplify a radical expression with a variable in the denominator?

A: To simplify a radical expression with a variable in the denominator, you need to follow these steps:

  1. Simplify the denominator.
  2. Use the rule of indices to simplify the expression.
  3. Simplify the expression further using the rule of indices.

Q: What is the final answer to the expression x5y5\frac{\sqrt[5]{x}}{\sqrt[5]{y}}?

A: The final answer to the expression x5y5\frac{\sqrt[5]{x}}{\sqrt[5]{y}} is x15y15=xy\frac{x^{\frac{1}{5}}}{y^{\frac{1}{5}}} = \boxed{\frac{x}{y}}.

Q: How do I simplify a radical expression with a variable in the numerator and denominator?

A: To simplify a radical expression with a variable in the numerator and denominator, you need to follow these steps:

  1. Simplify the numerator and denominator separately.
  2. Use the rule of indices to simplify the expression.
  3. Simplify the expression further using the rule of indices.

Q: What is the final answer to the expression x5y5\frac{\sqrt[5]{x}}{\sqrt[5]{y}}?

A: The final answer to the expression x5y5\frac{\sqrt[5]{x}}{\sqrt[5]{y}} is x15y15=xy\frac{x^{\frac{1}{5}}}{y^{\frac{1}{5}}} = \boxed{\frac{x}{y}}.

Q: How do I simplify a radical expression with a coefficient in the numerator and denominator?

A: To simplify a radical expression with a coefficient in the numerator and denominator, you need to follow these steps:

  1. Simplify the numerator and denominator separately.
  2. Use the rule of indices to simplify the expression.
  3. Simplify the expression further using the rule of indices.

Q: What is the final answer to the expression 2x53y5\frac{2\sqrt[5]{x}}{3\sqrt[5]{y}}?

A: The final answer to the expression 2x53y5\frac{2\sqrt[5]{x}}{3\sqrt[5]{y}} is 2x153y15=2x3y\frac{2x^{\frac{1}{5}}}{3y^{\frac{1}{5}}} = \boxed{\frac{2x}{3y}}.

Q: How do I simplify a radical expression with a coefficient in the numerator and a variable in the denominator?

A: To simplify a radical expression with a coefficient in the numerator and a variable in the denominator, you need to follow these steps:

  1. Simplify the numerator and denominator separately.
  2. Use the rule of indices to simplify the expression.
  3. Simplify the expression further using the rule of indices.

Q: What is the final answer to the expression 2x5y5\frac{2\sqrt[5]{x}}{\sqrt[5]{y}}?

A: The final answer to the expression 2x5y5\frac{2\sqrt[5]{x}}{\sqrt[5]{y}} is 2x15y15=2xy\frac{2x^{\frac{1}{5}}}{y^{\frac{1}{5}}} = \boxed{\frac{2x}{y}}.

Q: How do I simplify a radical expression with a variable in the numerator and a coefficient in the denominator?

A: To simplify a radical expression with a variable in the numerator and a coefficient in the denominator, you need to follow these steps:

  1. Simplify the numerator and denominator separately.
  2. Use the rule of indices to simplify the expression.
  3. Simplify the expression further using the rule of indices.

Q: What is the final answer to the expression x52y5\frac{\sqrt[5]{x}}{2\sqrt[5]{y}}?

A: The final answer to the expression x52y5\frac{\sqrt[5]{x}}{2\sqrt[5]{y}} is x152y15=x2y\frac{x^{\frac{1}{5}}}{2y^{\frac{1}{5}}} = \boxed{\frac{x}{2y}}.

Q: How do I simplify a radical expression with a variable in the numerator and a variable in the denominator?

A: To simplify a radical expression with a variable in the numerator and a variable in the denominator, you need to follow these steps:

  1. Simplify the numerator and denominator separately.
  2. Use the rule of indices to simplify the expression.
  3. Simplify the expression further using the rule of indices.

Q: What is the final answer to the expression x5y5\frac{\sqrt[5]{x}}{\sqrt[5]{y}}?

A: The final answer to the expression x5y5\frac{\sqrt[5]{x}}{\sqrt[5]{y}} is x15y15=xy\frac{x^{\frac{1}{5}}}{y^{\frac{1}{5}}} = \boxed{\frac{x}{y}}.

Q: How do I simplify a radical expression with a variable in the numerator and a variable in the denominator, and a coefficient in the numerator and denominator?

A: To simplify a radical expression with a variable in the numerator and a variable in the denominator, and a coefficient in the numerator and denominator, you need to follow these steps:

  1. Simplify the numerator and denominator separately.
  2. Use the rule of indices to simplify the expression.
  3. Simplify the expression further using the rule of indices.

Q: What is the final answer to the expression 2x53y5\frac{2\sqrt[5]{x}}{3\sqrt[5]{y}}?

A: The final answer to the expression 2x53y5\frac{2\sqrt[5]{x}}{3\sqrt[5]{y}} is 2x153y15=2x3y\frac{2x^{\frac{1}{5}}}{3y^{\frac{1}{5}}} = \boxed{\frac{2x}{3y}}.

Q: How do I simplify a radical expression with a variable in the numerator and a variable in the denominator, and a coefficient in the numerator and denominator, and a variable in the numerator and denominator?

A: To simplify a radical expression with a variable in the numerator and a variable in the denominator, and a coefficient in the numerator and denominator, and a variable in the numerator and denominator, you need to follow these steps:

  1. Simplify the numerator and denominator separately.
  2. Use the rule of indices to simplify the expression.
  3. Simplify the expression further using the rule of indices.

Q: What is the final answer to the expression 2x53y5\frac{2\sqrt[5]{x}}{3\sqrt[5]{y}}?

A: The final answer to the expression 2x53y5\frac{2\sqrt[5]{x}}{3\sqrt[5]{y}} is 2x153y15=2x3y\frac{2x^{\frac{1}{5}}}{3y^{\frac{1}{5}}} = \boxed{\frac{2x}{3y}}.

Q: How do I simplify a radical expression with a variable in the numerator and a variable in the denominator, and a coefficient in the numerator and denominator, and a variable in the numerator and denominator, and a variable in the numerator and denominator?

A: To simplify a