Which Of The Following Is Equivalent To $\sqrt[5]{13^3}$?A. $13^2$B. $13^{15}$C. $13^{\frac{5}{3}}$D. $13^{\frac{3}{5}}$

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the concept of radical expressions, specifically focusing on the expression $\sqrt[5]{13^3}$, and determine which of the given options is equivalent to it.

Understanding Radical Expressions

A radical expression is a mathematical expression that involves a root or a power of a number. The most common radical expressions are square roots, cube roots, and nth roots. The general form of a radical expression is $\sqrt[n]{a}$, where $a$ is the radicand and $n$ is the index of the root.

Simplifying the Expression $\sqrt[5]{13^3}$

To simplify the expression $\sqrt[5]{13^3}$, we need to understand the properties of exponents and roots. The expression can be rewritten as $(13^3)^{\frac{1}{5}}$. Using the property of exponents that states $(a^m)^n = a^{mn}$, we can simplify the expression further.

Applying the Property of Exponents

Using the property of exponents, we can rewrite the expression as $13^{3 \times \frac{1}{5}}$. Simplifying the exponent, we get $13^{\frac{3}{5}}$.

Evaluating the Options

Now that we have simplified the expression $\sqrt[5]{13^3}$ to $13^{\frac{3}{5}}$, let's evaluate the options given:

A. $13^2$ B. $13^{15}$ C. $13^{\frac{5}{3}}$ D. $13^{\frac{3}{5}}$

Option A: $13^2$

Option A is not equivalent to the simplified expression $13^{\frac{3}{5}}$. The exponent $2$ is not equal to $\frac{3}{5}$, so this option is incorrect.

Option B: $13^{15}$

Option B is also not equivalent to the simplified expression $13^{\frac{3}{5}}$. The exponent $15$ is not equal to $\frac{3}{5}$, so this option is incorrect.

Option C: $13^{\frac{5}{3}}$

Option C is not equivalent to the simplified expression $13^{\frac{3}{5}}$. The exponent $\frac{5}{3}$ is not equal to $\frac{3}{5}$, so this option is incorrect.

Option D: $13^{\frac{3}{5}}$

Option D is equivalent to the simplified expression $13^{\frac{3}{5}}$. The exponent $\frac{3}{5}$ is equal to the exponent in the simplified expression, so this option is correct.

Conclusion

In conclusion, the expression $\sqrt[5]{13^3}$ is equivalent to $13^{\frac{3}{5}}$. This is because the property of exponents allows us to simplify the expression by rewriting it as $13^{3 \times \frac{1}{5}}$, which simplifies to $13^{\frac{3}{5}}$. Therefore, the correct answer is option D, $13^{\frac{3}{5}}$.

Additional Tips and Tricks

When simplifying radical expressions, it's essential to understand the properties of exponents and roots. By applying these properties, you can simplify complex expressions and arrive at the correct solution. Additionally, make sure to evaluate the options carefully and check if they are equivalent to the simplified expression.

Common Mistakes to Avoid

When simplifying radical expressions, some common mistakes to avoid include:

• Not applying the property of exponents correctly
• Not simplifying the expression fully
• Not evaluating the options carefully

By avoiding these common mistakes, you can ensure that you arrive at the correct solution and simplify radical expressions with confidence.

Final Thoughts

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In our previous article, we explored the concept of radical expressions and simplified the expression $\sqrt[5]{13^3}$ to $13^{\frac{3}{5}}$. In this article, we will answer some frequently asked questions about simplifying radical expressions.

Q&A

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

A: A radical expression is a mathematical expression that involves a root or a power of a number, while an exponential expression is a mathematical expression that involves a power of a number. For example, $\sqrt[5]{13^3}$ is a radical expression, while $13^3$ is an exponential expression.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to understand the properties of exponents and roots. You can use the property of exponents that states $(a^m)^n = a^{mn}$ to simplify the expression. For example, $\sqrt[5]{13^3}$ can be rewritten as $(13^3)^{\frac{1}{5}}$, which simplifies to $13^{\frac{3}{5}}$.

Q: What is the index of a radical expression?

A: The index of a radical expression is the number that is outside the radical sign. For example, in the expression $\sqrt[5]{13^3}$, the index is 5.

Q: What is the radicand of a radical expression?

A: The radicand of a radical expression is the number that is inside the radical sign. For example, in the expression $\sqrt[5]{13^3}$, the radicand is $13^3$.

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

A: Yes, you can simplify a radical expression with a negative exponent. For example, $\sqrt[5]{13^{-3}}$ can be rewritten as $(13^{-3})^{\frac{1}{5}}$, which simplifies to $13^{-\frac{3}{5}}$.

Q: How do I evaluate the options when simplifying a radical expression?

A: When evaluating the options, make sure to check if they are equivalent to the simplified expression. You can do this by applying the property of exponents and simplifying the expression further. For example, if the simplified expression is $13^{\frac{3}{5}}$, you can check if the options are equivalent by rewriting them as powers of 13.

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

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

• Not applying the property of exponents correctly
• Not simplifying the expression fully
• Not evaluating the options carefully

By avoiding these common mistakes, you can ensure that you arrive at the correct solution and simplify radical expressions with confidence.

Additional Tips and Tricks

When simplifying radical expressions, it's essential to understand the properties of exponents and roots. By applying these properties, you can simplify complex expressions and arrive at the correct solution. Additionally, make sure to evaluate the options carefully and check if they are equivalent to the simplified expression.

Conclusion

Simplifying radical expressions is a crucial skill in mathematics, and understanding the properties of exponents and roots is essential for success. By applying the property of exponents and simplifying the expression $\sqrt[5]{13^3}$, we arrived at the correct solution, $13^{\frac{3}{5}}$. Remember to evaluate the options carefully and check if they are equivalent to the simplified expression. With practice and patience, you can become proficient in simplifying radical expressions and tackle complex mathematical problems with confidence.

Final Thoughts

Simplifying radical expressions is a fundamental concept in mathematics, and understanding the properties of exponents and roots is essential for success. By applying the property of exponents and simplifying the expression $\sqrt[5]{13^3}$, we arrived at the correct solution, $13^{\frac{3}{5}}$. Remember to evaluate the options carefully and check if they are equivalent to the simplified expression. With practice and patience, you can become proficient in simplifying radical expressions and tackle complex mathematical problems with confidence.