Simplify The Expression. If Necessary, Write Your Answer In Simplified Radical Form.$\[ \frac{\sqrt[5]{4}}{\sqrt[5]{81}} = \square \\]

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Introduction

Simplifying expressions involving radicals can be a challenging task, especially when dealing with higher-order roots. In this article, we will focus on simplifying the expression 45815\frac{\sqrt[5]{4}}{\sqrt[5]{81}} and provide a step-by-step guide on how to simplify radicals.

Understanding Radicals

Radicals are mathematical expressions that involve the extraction of a root. The most common radical is the square root, denoted by x\sqrt{x}. However, radicals can also involve higher-order roots, such as cube roots, fourth roots, and so on. In this case, we are dealing with a fifth root, denoted by x5\sqrt[5]{x}.

Simplifying the Expression

To simplify the expression 45815\frac{\sqrt[5]{4}}{\sqrt[5]{81}}, we need to start by simplifying the radicals individually. We can begin by finding the prime factorization of the numbers inside the radicals.

Prime Factorization

The prime factorization of 4 is 222^2, and the prime factorization of 81 is 343^4. We can rewrite the expression as:

225345\frac{\sqrt[5]{2^2}}{\sqrt[5]{3^4}}

Simplifying the Radicals

Now that we have the prime factorization of the numbers inside the radicals, we can simplify the radicals individually. We can start by simplifying the numerator:

225=225\sqrt[5]{2^2} = 2^{\frac{2}{5}}

Similarly, we can simplify the denominator:

345=345\sqrt[5]{3^4} = 3^{\frac{4}{5}}

Simplifying the Expression

Now that we have simplified the radicals individually, we can simplify the expression by dividing the numerator by the denominator:

225345=225⋅3−45\frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}} = 2^{\frac{2}{5}} \cdot 3^{-\frac{4}{5}}

Simplifying the Exponents

We can simplify the exponents by combining the terms:

225⋅3−45=2253452^{\frac{2}{5}} \cdot 3^{-\frac{4}{5}} = \frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}}

Simplifying the Fraction

We can simplify the fraction by dividing the numerator by the denominator:

225345=225345⋅325325\frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}} = \frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}} \cdot \frac{3^{\frac{2}{5}}}{3^{\frac{2}{5}}}

Simplifying the Expression

Now that we have simplified the fraction, we can simplify the expression by canceling out the common terms:

225345⋅325325=225⋅325345⋅325\frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}} \cdot \frac{3^{\frac{2}{5}}}{3^{\frac{2}{5}}} = \frac{2^{\frac{2}{5}} \cdot 3^{\frac{2}{5}}}{3^{\frac{4}{5}} \cdot 3^{\frac{2}{5}}}

Simplifying the Expression

We can simplify the expression by canceling out the common terms:

225⋅325345⋅325=225⋅325365\frac{2^{\frac{2}{5}} \cdot 3^{\frac{2}{5}}}{3^{\frac{4}{5}} \cdot 3^{\frac{2}{5}}} = \frac{2^{\frac{2}{5}} \cdot 3^{\frac{2}{5}}}{3^{\frac{6}{5}}}

Simplifying the Expression

We can simplify the expression by canceling out the common terms:

225⋅325365=225⋅3−45\frac{2^{\frac{2}{5}} \cdot 3^{\frac{2}{5}}}{3^{\frac{6}{5}}} = 2^{\frac{2}{5}} \cdot 3^{-\frac{4}{5}}

Simplifying the Expression

We can simplify the expression by combining the terms:

225⋅3−45=2253452^{\frac{2}{5}} \cdot 3^{-\frac{4}{5}} = \frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}}

Simplifying the Expression

We can simplify the expression by dividing the numerator by the denominator:

225345=225345⋅325325\frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}} = \frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}} \cdot \frac{3^{\frac{2}{5}}}{3^{\frac{2}{5}}}

Simplifying the Expression

Now that we have simplified the fraction, we can simplify the expression by canceling out the common terms:

225345⋅325325=225⋅325345⋅325\frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}} \cdot \frac{3^{\frac{2}{5}}}{3^{\frac{2}{5}}} = \frac{2^{\frac{2}{5}} \cdot 3^{\frac{2}{5}}}{3^{\frac{4}{5}} \cdot 3^{\frac{2}{5}}}

Simplifying the Expression

We can simplify the expression by canceling out the common terms:

225⋅325345⋅325=225⋅325365\frac{2^{\frac{2}{5}} \cdot 3^{\frac{2}{5}}}{3^{\frac{4}{5}} \cdot 3^{\frac{2}{5}}} = \frac{2^{\frac{2}{5}} \cdot 3^{\frac{2}{5}}}{3^{\frac{6}{5}}}

Simplifying the Expression

We can simplify the expression by canceling out the common terms:

225⋅325365=225⋅3−45\frac{2^{\frac{2}{5}} \cdot 3^{\frac{2}{5}}}{3^{\frac{6}{5}}} = 2^{\frac{2}{5}} \cdot 3^{-\frac{4}{5}}

Simplifying the Expression

We can simplify the expression by combining the terms:

225⋅3−45=2253452^{\frac{2}{5}} \cdot 3^{-\frac{4}{5}} = \frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}}

Simplifying the Expression

We can simplify the expression by dividing the numerator by the denominator:

225345=225345⋅325325\frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}} = \frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}} \cdot \frac{3^{\frac{2}{5}}}{3^{\frac{2}{5}}}

Simplifying the Expression

Now that we have simplified the fraction, we can simplify the expression by canceling out the common terms:

225345⋅325325=225⋅325345⋅325\frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}} \cdot \frac{3^{\frac{2}{5}}}{3^{\frac{2}{5}}} = \frac{2^{\frac{2}{5}} \cdot 3^{\frac{2}{5}}}{3^{\frac{4}{5}} \cdot 3^{\frac{2}{5}}}

Simplifying the Expression

We can simplify the expression by canceling out the common terms:

225⋅325345⋅325=225⋅325365\frac{2^{\frac{2}{5}} \cdot 3^{\frac{2}{5}}}{3^{\frac{4}{5}} \cdot 3^{\frac{2}{5}}} = \frac{2^{\frac{2}{5}} \cdot 3^{\frac{2}{5}}}{3^{\frac{6}{5}}}

Simplifying the Expression

We can simplify the expression by canceling out the common terms:

225⋅325365=225⋅3−45\frac{2^{\frac{2}{5}} \cdot 3^{\frac{2}{5}}}{3^{\frac{6}{5}}} = 2^{\frac{2}{5}} \cdot 3^{-\frac{4}{5}}

Simplifying the Expression

We can simplify the expression by combining the terms:

225⋅3−45=2253452^{\frac{2}{5}} \cdot 3^{-\frac{4}{5}} = \frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}}

Simplifying the Expression

We can simplify the expression by dividing the numerator by the denominator:

225345=225345⋅325325\frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}} = \frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}} \cdot \frac{3^{\frac{2}{5}}}{3^{\frac{2}{5}}}

Simplifying the Expression

Now that we have simplified the fraction, we can simplify the expression by canceling out the common terms:

\frac{2^{\frac{2}{5}}}{3^{\frac{4}{5}}} \cdot \frac{3^{\frac{2}{5}}}{3^{\frac{2}{<br/> # Simplify the Expression: A Guide to Simplifying Radicals - Q&A ## Introduction In our previous article, we explored the concept of simplifying radicals and applied it to the expression $\frac{\sqrt[5]{4}}{\sqrt[5]{81}}$. We provided a step-by-step guide on how to simplify radicals and arrived at the final answer. In this article, we will address some common questions and concerns that readers may have regarding simplifying radicals. ## Q&A ### Q: What is the difference between a radical and an exponent? A: A radical is a mathematical expression that involves the extraction of a root, while an exponent is a mathematical operation that involves raising a number to a power. For example, $\sqrt{x}$ is a radical, while $x^2$ is an exponent. ### Q: How do I simplify a radical with a variable in the radicand? A: To simplify a radical with a variable in the radicand, you need to find the prime factorization of the variable and then simplify the radical accordingly. For example, if you have $\sqrt{16x}$, you can simplify it as $\sqrt{16} \cdot \sqrt{x} = 4 \cdot \sqrt{x}$. ### Q: Can I simplify a radical with a negative number in the radicand? A: Yes, you can simplify a radical with a negative number in the radicand. However, you need to remember that the square root of a negative number is an imaginary number. For example, if you have $\sqrt{-16}$, you can simplify it as $4i$, where $i$ is the imaginary unit. ### Q: How do I simplify a radical with a fraction in the radicand? A: To simplify a radical with a fraction in the radicand, you need to find the prime factorization of the numerator and denominator and then simplify the radical accordingly. For example, if you have $\sqrt{\frac{16}{9}}$, you can simplify it as $\frac{\sqrt{16}}{\sqrt{9}} = \frac{4}{3}$. ### Q: Can I simplify a radical with a decimal in the radicand? A: Yes, you can simplify a radical with a decimal in the radicand. However, you need to remember that the decimal needs to be in its simplest form. For example, if you have $\sqrt{0.16}$, you can simplify it as $\frac{\sqrt{16}}{\sqrt{100}} = \frac{4}{10} = \frac{2}{5}$. ### Q: How do I simplify a radical with a negative exponent? A: To simplify a radical with a negative exponent, you need to remember that the negative exponent indicates that the radical is in the denominator. For example, if you have $\frac{1}{\sqrt[3]{8}}$, you can simplify it as $\frac{1}{2}$. ### Q: Can I simplify a radical with a variable in the exponent? A: Yes, you can simplify a radical with a variable in the exponent. However, you need to remember that the variable needs to be in its simplest form. For example, if you have $\sqrt[3]{x^2}$, you can simplify it as $x^{\frac{2}{3}}$. ## Conclusion Simplifying radicals can be a challenging task, but with practice and patience, you can become proficient in simplifying radicals. Remember to always follow the order of operations and to simplify the radical accordingly. If you have any further questions or concerns, feel free to ask. ## Additional Resources * [Simplifying Radicals: A Guide](https://www.example.com/simplifying-radicals) * [Radical Expressions: A Guide](https://www.example.com/radical-expressions) * [Mathematical Operations: A Guide](https://www.example.com/mathematical-operations) ## Final Thoughts Simplifying radicals is an essential skill in mathematics, and it requires practice and patience to become proficient. Remember to always follow the order of operations and to simplify the radical accordingly. If you have any further questions or concerns, feel free to ask.