Given $x\ \textgreater \ 0$ And $y\ \textgreater \ 0$, Select The Expression That Is Equivalent To $\sqrt[4]{81 X^6 Y^8}$.Answer:A. $9 X^{\frac{2}{3}} Y^{\frac{1}{2}}$B. \$9 X^{\frac{3}{2}}
Simplifying Radical Expressions: A Step-by-Step Guide
Radical expressions can be complex and challenging to simplify, but with the right techniques and strategies, they can be broken down into more manageable parts. In this article, we will explore how to simplify radical expressions, focusing on the given problem of finding an equivalent expression to $\sqrt[4]{81 x^6 y^8}$.
The problem requires us to simplify the radical expression $\sqrt[4]{81 x^6 y^8}$, given that $x > 0$ and $y > 0$. To simplify this expression, we need to apply the properties of radicals and exponents.
Breaking Down the Radical Expression
The given radical expression can be broken down into two parts: the radicand and the index. The radicand is $81 x^6 y^8$, and the index is $4$.
Simplifying the Radicand
To simplify the radicand, we need to find the prime factorization of $81 x^6 y^8$. We can start by breaking down $81$ into its prime factors:
Now, we can rewrite the radicand as:
Applying the Properties of Radicals
Now that we have the prime factorization of the radicand, we can apply the properties of radicals to simplify the expression. One of the properties of radicals is that we can take the $n$th root of a number by dividing the exponent by $n$.
In this case, we have a 4th root, so we can divide the exponents by $4$:
Simplifying the Exponents
Now that we have divided the exponents by $4$, we can simplify them further. We can rewrite the exponents as:
In conclusion, the expression $\sqrt[4]{81 x^6 y^8}$ is equivalent to $3 x^{\frac{3}{2}} y^2$. This is the correct answer, which is option B.
Why is Option A Incorrect?
Option A is incorrect because it does not accurately simplify the radical expression. The expression $9 x^{\frac{2}{3}} y^{\frac{1}{2}}$ does not have the same exponents as the simplified expression $3 x^{\frac{3}{2}} y^2$.
Why is Option C Incorrect?
Option C is incorrect because it does not accurately simplify the radical expression. The expression $9 x^{\frac{3}{2}} y^{\frac{1}{2}}$ does not have the same exponents as the simplified expression $3 x^{\frac{3}{2}} y^2$.
Simplifying radical expressions can be challenging, but with the right techniques and strategies, it can be done. By breaking down the radical expression into its components, applying the properties of radicals, and simplifying the exponents, we can arrive at the correct answer. In this article, we have explored how to simplify the radical expression $\sqrt[4]{81 x^6 y^8}$, and we have seen that the correct answer is option B.
If you are struggling with simplifying radical expressions, there are many additional resources available to help you. Some of these resources include:
- Online tutorials and videos
- Math textbooks and workbooks
- Online communities and forums
- Math tutors and instructors
By taking advantage of these resources, you can improve your skills and become more confident in your ability to simplify radical expressions.
In conclusion, simplifying radical expressions is an important skill that can be applied to a wide range of mathematical problems. By breaking down the radical expression into its components, applying the properties of radicals, and simplifying the exponents, we can arrive at the correct answer. In this article, we have explored how to simplify the radical expression $\sqrt[4]{81 x^6 y^8}$, and we have seen that the correct answer is option B.
Simplifying Radical Expressions: A Q&A Guide
In our previous article, we explored how to simplify radical expressions, focusing on the given problem of finding an equivalent expression to $\sqrt[4]{81 x^6 y^8}$. In this article, we will continue to provide a Q&A guide to help you better understand and apply the concepts of simplifying radical expressions.
Q: What is a radical expression?
A: A radical expression is an expression that contains a root, such as a square root, cube root, or fourth root. Radical expressions can be written in the form $\sqrt[n]{a}$, where $n$ is the index of the root and $a$ is the radicand.
Q: What is the index of a radical expression?
A: The index of a radical expression is the number that is written outside the radical sign. For example, in the expression $\sqrt[4]{a}$, the index is $4$.
Q: What is the radicand of a radical expression?
A: The radicand of a radical expression is the number or expression that is written inside the radical sign. For example, in the expression $\sqrt[4]{a}$, the radicand is $a$.
Q: How do I simplify a radical expression?
A: To simplify a radical expression, you need to follow these steps:
- Break down the radicand into its prime factors.
- Apply the properties of radicals to simplify the expression.
- Simplify the exponents.
Q: What are the properties of radicals?
A: The properties of radicals are:
- The product of two radicals is equal to the radical of the product of the two numbers.
- The quotient of two radicals is equal to the radical of the quotient of the two numbers.
- The power of a radical is equal to the radical of the power of the number.
Q: How do I apply the properties of radicals to simplify an expression?
A: To apply the properties of radicals, you need to follow these steps:
- Identify the properties that can be applied to the expression.
- Apply the properties to simplify the expression.
- Simplify the exponents.
Q: What are some common mistakes to avoid when simplifying radical expressions?
A: Some common mistakes to avoid when simplifying radical expressions include:
- Not breaking down the radicand into its prime factors.
- Not applying the properties of radicals correctly.
- Not simplifying the exponents.
Q: How do I check my work when simplifying radical expressions?
A: To check your work when simplifying radical expressions, you need to follow these steps:
- Plug in values for the variables.
- Simplify the expression.
- Check if the simplified expression is equivalent to the original expression.
Q: What are some real-world applications of simplifying radical expressions?
A: Simplifying radical expressions has many real-world applications, including:
- Calculating distances and heights in geometry and trigonometry.
- Solving equations and inequalities in algebra.
- Working with complex numbers and polynomials.
In conclusion, simplifying radical expressions is an important skill that can be applied to a wide range of mathematical problems. By following the steps outlined in this article, you can become more confident in your ability to simplify radical expressions and apply them to real-world problems.
If you are struggling with simplifying radical expressions, there are many additional resources available to help you. Some of these resources include:
- Online tutorials and videos
- Math textbooks and workbooks
- Online communities and forums
- Math tutors and instructors
By taking advantage of these resources, you can improve your skills and become more confident in your ability to simplify radical expressions.
Simplifying radical expressions is a challenging but rewarding skill that can be applied to a wide range of mathematical problems. By following the steps outlined in this article and practicing regularly, you can become more confident in your ability to simplify radical expressions and apply them to real-world problems.