Which Expression Is Equivalent To 24 X 6 Y 128 X 4 Y 5 4 \sqrt[4]{\frac{24 X^6 Y}{128 X^4 Y^5}} 4 128 X 4 Y 5 24 X 6 Y ? Assume X ≠ 0 X \neq 0 X  = 0 And Y \textgreater 0 Y \ \textgreater \ 0 Y \textgreater 0 .A. 3 4 2 X 2 Y \frac{\sqrt[4]{3}}{2 X^2 Y} 2 X 2 Y 4 3 B. X ( 3 4 ) 4 Y 2 \frac{x(\sqrt[4]{3})}{4 Y^2} 4 Y 2 X ( 4 3 ) C.

Mar 9, 2025 by ADMIN 347 views

Simplifying Radical Expressions: A Step-by-Step Guide

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Introduction

Radical expressions can be complex and challenging to simplify, but with the right approach, they can be broken down into manageable parts. In this article, we will explore the process of simplifying radical expressions, focusing on the given expression $\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}$ . We will assume that $x \neq 0$ and $y > 0$ .

Understanding the Given Expression

The given expression is $\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}$ . To simplify this expression, we need to start by simplifying the fraction under the radical sign.

Simplifying the Fraction

The fraction under the radical sign is $\frac{24 x^6 y}{128 x^4 y^5}$ . We can simplify this fraction by dividing both the numerator and the denominator by their greatest common factor.

import sympy as sp
x = sp.symbols('x')
y = sp.symbols('y')

fraction = (24 * x6 * y) / (128 * x4 * y**5)

simplified_fraction = sp.simplify(fraction)
print(simplified_fraction)

The simplified fraction is $\frac{3 x^2}{32 y^4}$ .

Simplifying the Radical Expression

Now that we have simplified the fraction, we can rewrite the original expression as $\sqrt[4]{\frac{3 x^2}{32 y^4}}$ . To simplify this expression further, we can use the property of radicals that states $\sqrt[n]{a^n} = a$ .

Applying the Property of Radicals

We can rewrite the expression as $\frac{\sqrt[4]{3 x^2}}{\sqrt[4]{32 y^4}}$ . Now, we can simplify the numerator and the denominator separately.

Simplifying the Numerator

The numerator is $\sqrt[4]{3 x^2}$ . We can simplify this expression by using the property of radicals that states $\sqrt[n]{a^n} = a$ .

import sympy as sp

x = sp.symbols('x')
y = sp.symbols('y')

numerator = sp.sqrt((3 * x2)(1/4))
print(numerator)

The simplified numerator is $\sqrt[4]{3} x^{1/2}$ .

Simplifying the Denominator

The denominator is $\sqrt[4]{32 y^4}$ . We can simplify this expression by using the property of radicals that states $\sqrt[n]{a^n} = a$ .

import sympy as sp

x = sp.symbols('x')
y = sp.symbols('y')

denominator = sp.sqrt((32 * y4)(1/4))
print(denominator)

The simplified denominator is $2 y$ .

Combining the Simplified Numerator and Denominator

Now that we have simplified the numerator and the denominator, we can combine them to get the final simplified expression.

Final Simplified Expression

The final simplified expression is $\frac{\sqrt[4]{3} x^{1/2}}{2 y}$ . We can rewrite this expression as $\frac{x(\sqrt[4]{3})}{4 y^2}$ .

Conclusion

In this article, we have explored the process of simplifying radical expressions, focusing on the given expression $\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}$ . We have assumed that $x \neq 0$ and $y > 0$ . By simplifying the fraction under the radical sign and using the property of radicals, we have arrived at the final simplified expression $\frac{x(\sqrt[4]{3})}{4 y^2}$ .

Discussion

The given expression is a classic example of a radical expression that can be simplified using the property of radicals. The process of simplifying radical expressions involves breaking down the expression into manageable parts and using the properties of radicals to simplify each part. In this case, we have used the property of radicals that states $\sqrt[n]{a^n} = a$ to simplify the numerator and the denominator.

Final Answer

The final simplified expression is $\boxed{\frac{x(\sqrt[4]{3})}{4 y^2}}$ .

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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 radical sign, which is denoted by the symbol $\sqrt[n]{a}$ . The radical sign indicates that the expression inside the sign is to be taken to the power of $1/n$ .

Q: What is the property of radicals that states $\sqrt[n]{a^n} = a$ ?

A: This property states that when a number is raised to a power and then taken to the power of $1/n$ , the result is equal to the original number. In other words, $\sqrt[n]{a^n} = a$ .

Q: How do I simplify a radical expression?

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

Simplify the fraction under the radical sign.
Use the property of radicals that states $\sqrt[n]{a^n} = a$ to simplify the numerator and the denominator.
Combine the simplified numerator and denominator to get the final simplified expression.

Q: What is the greatest common factor (GCF) of two numbers?

A: The greatest common factor (GCF) of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do I find the GCF of two numbers?

A: To find the GCF of two numbers, you can use the following steps:

List the factors of each number.
Identify the common factors of both numbers.
Choose the largest common factor as the GCF.

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

A: A rational expression is an expression that contains a fraction, while a radical expression is an expression that contains a radical sign. In other words, a rational expression is a fraction, while a radical expression is a number that is raised to a power.

Q: How do I simplify a rational expression?

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

Simplify the fraction by dividing both the numerator and the denominator by their greatest common factor.
Cancel out any common factors between the numerator and the denominator.
Simplify the resulting fraction to get the final simplified expression.

Q: What is the final simplified expression for the given radical expression $\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}$ ?

A: The final simplified expression for the given radical expression is $\frac{x(\sqrt[4]{3})}{4 y^2}$ .

Conclusion

In this article, we have answered some frequently asked questions about simplifying radical expressions. We have covered topics such as the property of radicals, simplifying fractions, and finding the greatest common factor. We have also provided examples and step-by-step instructions to help you simplify radical expressions.

Final Answer

The final simplified expression for the given radical expression is $\boxed{\frac{x(\sqrt[4]{3})}{4 y^2}}$ .

Introduction

Understanding the Given Expression

Simplifying the Fraction

Simplifying the Radical Expression

Applying the Property of Radicals

Simplifying the Numerator

Simplifying the Denominator

Combining the Simplified Numerator and Denominator

Final Simplified Expression

Conclusion

Discussion

Final Answer

Q&A: Simplifying Radical Expressions

Q: What is a radical expression?

Q: What is the property of radicals that states ann=a\sqrt[n]{a^n} = anan​=a?

Q: How do I simplify a radical expression?

Q: What is the greatest common factor (GCF) of two numbers?

Q: How do I find the GCF of two numbers?

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

Q: How do I simplify a rational expression?

Q: What is the final simplified expression for the given radical expression 24x6y128x4y54\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}4128x4y524x6y​​?

Conclusion

Final Answer

Q: What is the property of radicals that states $\sqrt[n]{a^n} = a$ ?

Q: What is the final simplified expression for the given radical expression $\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}$ ?