Rewriting Expressions With Rational Exponents Which Expression Is Equivalent To 81 Superscript One-third? 3 RootIndex 3 StartRoot 3 EndRoot 3 RootIndex 3 StartRoot 3 Cubed EndRoot 9 RootIndex 3 StartRoot 3 EndRoot 27 RootIndex 3 StartRoot 3 EndRoot

Understanding Rational Exponents

Rational exponents are a way to express roots and powers of numbers using fractions. They are a powerful tool in mathematics, particularly in algebra and calculus. In this article, we will explore how to rewrite expressions with rational exponents and apply this knowledge to solve a specific problem.

What are Rational Exponents?

Rational exponents are expressions of the form $a^{\frac{m}{n}}$, where $a$ is a real number and $m$ and $n$ are integers. The exponent $\frac{m}{n}$ is called a rational exponent. Rational exponents can be used to simplify expressions involving roots and powers.

Rewriting Expressions with Rational Exponents

To rewrite an expression with a rational exponent, we can use the following formula:

$a^{\frac{m}{n}} = \sqrt[n]{a^m}$

This formula allows us to rewrite an expression with a rational exponent as a radical expression.

Applying Rational Exponents to the Problem

Now, let's apply this knowledge to the problem at hand. We are given the expression $81^{\frac{1}{3}}$ and asked to find an equivalent expression.

Step 1: Rewrite the Expression with a Rational Exponent

Using the formula above, we can rewrite the expression $81^{\frac{1}{3}}$ as:

$\sqrt[3]{81^1}$

Step 2: Simplify the Expression

Now, we can simplify the expression by evaluating the exponent:

$\sqrt[3]{81^1} = \sqrt[3]{81}$

Step 3: Evaluate the Radical

Finally, we can evaluate the radical by finding the cube root of 81:

$\sqrt[3]{81} = 3\sqrt[3]{3}$

Conclusion

In this article, we have learned how to rewrite expressions with rational exponents and applied this knowledge to solve a specific problem. We have seen how to use the formula $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ to rewrite an expression with a rational exponent as a radical expression. We have also seen how to simplify and evaluate the resulting radical expression.

The Final Answer

So, which expression is equivalent to $81^{\frac{1}{3}}$?

The correct answer is:

3 RootIndex 3 StartRoot 3 EndRoot

This expression is equivalent to $81^{\frac{1}{3}}$ because it can be rewritten as:

$\sqrt[3]{81} = 3\sqrt[3]{3}$

Discussion

Rational exponents are a powerful tool in mathematics, and understanding how to rewrite expressions with rational exponents is essential for solving problems involving roots and powers. In this article, we have seen how to use the formula $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ to rewrite an expression with a rational exponent as a radical expression. We have also seen how to simplify and evaluate the resulting radical expression.

Common Mistakes

When working with rational exponents, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not using the correct formula: Make sure to use the formula $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ to rewrite an expression with a rational exponent.
• Not simplifying the expression: Make sure to simplify the expression by evaluating the exponent and evaluating the radical.
• Not evaluating the radical: Make sure to evaluate the radical by finding the cube root of the number.

Conclusion

In conclusion, rewriting expressions with rational exponents is a powerful tool in mathematics. By understanding how to use the formula $a^{\frac{m}{n}} = \sqrt[n]{a^m}$, we can rewrite expressions with rational exponents as radical expressions. We can then simplify and evaluate the resulting radical expression to find the final answer.

Final Answer

So, which expression is equivalent to $81^{\frac{1}{3}}$?

The correct answer is:

3 RootIndex 3 StartRoot 3 EndRoot

This expression is equivalent to $81^{\frac{1}{3}}$ because it can be rewritten as:

$\sqrt[3]{81} = 3\sqrt[3]{3}$

References

• [1] "Rational Exponents" by Math Open Reference
• [2] "Rewriting Expressions with Rational Exponents" by Khan Academy
• [3] "Rational Exponents and Radical Expressions" by Purplemath

Additional Resources

• [1] "Rational Exponents" by Wolfram MathWorld
• [2] "Rewriting Expressions with Rational Exponents" by MIT OpenCourseWare
• [3] "Rational Exponents and Radical Expressions" by IXL Math

FAQs

• Q: What is a rational exponent? A: A rational exponent is an expression of the form $a^{\frac{m}{n}}$, where $a$ is a real number and $m$ and $n$ are integers.
• Q: How do I rewrite an expression with a rational exponent? A: To rewrite an expression with a rational exponent, use the formula $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.
• Q: How do I simplify an expression with a rational exponent? A: To simplify an expression with a rational exponent, evaluate the exponent and evaluate the radical.

Conclusion

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Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about rewriting expressions with rational exponents.

Q: What is a rational exponent?

A: A rational exponent is an expression of the form $a^{\frac{m}{n}}$, where $a$ is a real number and $m$ and $n$ are integers.

Q: How do I rewrite an expression with a rational exponent?

A: To rewrite an expression with a rational exponent, use the formula $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

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

A: To simplify an expression with a rational exponent, evaluate the exponent and evaluate the radical.

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

A: A rational exponent is an expression of the form $a^{\frac{m}{n}}$, while a radical expression is an expression of the form $\sqrt[n]{a^m}$. While they are related, they are not the same thing.

Q: Can I use rational exponents to simplify radical expressions?

A: Yes, you can use rational exponents to simplify radical expressions. By rewriting a radical expression as a rational exponent, you can simplify it and make it easier to work with.

Q: How do I evaluate a rational exponent?

A: To evaluate a rational exponent, you need to evaluate the exponent and the radical separately. First, evaluate the exponent by raising the base to the power of the exponent. Then, evaluate the radical by finding the nth root of the result.

Q: Can I use rational exponents to solve equations involving radicals?

A: Yes, you can use rational exponents to solve equations involving radicals. By rewriting the radical expression as a rational exponent, you can simplify the equation and make it easier to solve.

Q: What are some common mistakes to avoid when working with rational exponents?

A: Some common mistakes to avoid when working with rational exponents include:

• Not using the correct formula to rewrite an expression with a rational exponent
• Not simplifying the expression by evaluating the exponent and the radical
• Not evaluating the radical by finding the nth root of the result
• Not checking the domain of the expression to ensure that it is defined

Q: How do I check the domain of a rational exponent?

A: To check the domain of a rational exponent, you need to ensure that the base is not equal to zero and that the exponent is a positive integer.

Q: Can I use rational exponents to solve problems involving complex numbers?

A: Yes, you can use rational exponents to solve problems involving complex numbers. By rewriting the complex number as a rational exponent, you can simplify the expression and make it easier to work with.

Q: What are some real-world applications of rational exponents?

A: Rational exponents have many real-world applications, including:

• Physics: Rational exponents are used to describe the motion of objects in terms of their position, velocity, and acceleration.
• Engineering: Rational exponents are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Computer Science: Rational exponents are used in algorithms and data structures to solve problems involving large datasets.

Conclusion

In conclusion, rational exponents are a powerful tool in mathematics that can be used to simplify and solve equations involving radicals. By understanding how to use rational exponents, you can solve a wide range of problems in mathematics and other fields. Remember to use the correct formula to rewrite an expression with a rational exponent, simplify the expression by evaluating the exponent and the radical, and check the domain of the expression to ensure that it is defined.