Simplify The Expression:${ \frac{\sqrt[4]{(x+y)^2} \sqrt[3]{x+y}}{\sqrt{(x+y)^3}} }$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. One of the common expressions that require simplification is the one involving radicals, such as the given expression: ${ \frac{\sqrt[4]{(x+y)^2} \sqrt[3]{x+y}}{\sqrt{(x+y)^3}} }$. In this article, we will delve into the world of radical expressions and provide a step-by-step guide on how to simplify this particular expression.

Understanding the Expression

Before we dive into the simplification process, let's break down the given expression and understand its components. The expression involves three radical terms:

  • (x+y)24\sqrt[4]{(x+y)^2}: This term represents the fourth root of the square of the sum of x and y.
  • x+y3\sqrt[3]{x+y}: This term represents the cube root of the sum of x and y.
  • (x+y)3\sqrt{(x+y)^3}: This term represents the square root of the cube of the sum of x and y.

Simplifying the Expression

To simplify the given expression, we need to apply the rules of exponents and radicals. Let's start by simplifying each radical term individually.

Simplifying the Fourth Root Term

The fourth root term can be simplified as follows:

(x+y)24=(x+y)24=(x+y)12=x+y\sqrt[4]{(x+y)^2} = (x+y)^{\frac{2}{4}} = (x+y)^{\frac{1}{2}} = \sqrt{x+y}

Simplifying the Cube Root Term

The cube root term can be simplified as follows:

x+y3=(x+y)13\sqrt[3]{x+y} = (x+y)^{\frac{1}{3}}

Simplifying the Square Root Term

The square root term can be simplified as follows:

(x+y)3=(x+y)32\sqrt{(x+y)^3} = (x+y)^{\frac{3}{2}}

Combining the Simplified Terms

Now that we have simplified each radical term, we can combine them to simplify the entire expression.

(x+y)24x+y3(x+y)3=x+yx+y3(x+y)32\frac{\sqrt[4]{(x+y)^2} \sqrt[3]{x+y}}{\sqrt{(x+y)^3}} = \frac{\sqrt{x+y} \sqrt[3]{x+y}}{(x+y)^{\frac{3}{2}}}

Rationalizing the Denominator

To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of (x+y)32(x+y)^{\frac{3}{2}} is (x+y)32(x+y)^{\frac{3}{2}}.

x+yx+y3(x+y)32⋅(x+y)32(x+y)32=x+yx+y3(x+y)32(x+y)3\frac{\sqrt{x+y} \sqrt[3]{x+y}}{(x+y)^{\frac{3}{2}}} \cdot \frac{(x+y)^{\frac{3}{2}}}{(x+y)^{\frac{3}{2}}} = \frac{\sqrt{x+y} \sqrt[3]{x+y} (x+y)^{\frac{3}{2}}}{(x+y)^3}

Final Simplification

Now that we have rationalized the denominator, we can simplify the expression further.

x+yx+y3(x+y)32(x+y)3=(x+y)12(x+y)13(x+y)32(x+y)3\frac{\sqrt{x+y} \sqrt[3]{x+y} (x+y)^{\frac{3}{2}}}{(x+y)^3} = \frac{(x+y)^{\frac{1}{2}} (x+y)^{\frac{1}{3}} (x+y)^{\frac{3}{2}}}{(x+y)^3}

Using the rule of exponents, we can combine the terms in the numerator:

(x+y)12(x+y)13(x+y)32(x+y)3=(x+y)12+13+32(x+y)3\frac{(x+y)^{\frac{1}{2}} (x+y)^{\frac{1}{3}} (x+y)^{\frac{3}{2}}}{(x+y)^3} = \frac{(x+y)^{\frac{1}{2} + \frac{1}{3} + \frac{3}{2}}}{(x+y)^3}

Simplifying the exponent in the numerator:

(x+y)12+13+32(x+y)3=(x+y)76(x+y)3\frac{(x+y)^{\frac{1}{2} + \frac{1}{3} + \frac{3}{2}}}{(x+y)^3} = \frac{(x+y)^{\frac{7}{6}}}{(x+y)^3}

Using the rule of exponents again, we can simplify the expression:

(x+y)76(x+y)3=(x+y)76−3\frac{(x+y)^{\frac{7}{6}}}{(x+y)^3} = (x+y)^{\frac{7}{6} - 3}

Simplifying the exponent:

(x+y)76−3=(x+y)−116(x+y)^{\frac{7}{6} - 3} = (x+y)^{-\frac{11}{6}}

Conclusion

In this article, we have simplified the given expression involving radicals. We started by breaking down the expression into its components and then applied the rules of exponents and radicals to simplify each term individually. Finally, we combined the simplified terms and rationalized the denominator to arrive at the final simplified expression.

Frequently Asked Questions

Q: What is the main concept behind simplifying radical expressions?

A: The main concept behind simplifying radical expressions is to apply the rules of exponents and radicals to simplify each term individually and then combine the simplified terms.

Q: How do I simplify a radical expression involving multiple terms?

A: To simplify a radical expression involving multiple terms, you need to apply the rules of exponents and radicals to simplify each term individually and then combine the simplified terms.

Q: What is the importance of rationalizing the denominator?

A: Rationalizing the denominator is important because it helps to eliminate any radicals in the denominator, making the expression easier to work with.

Q: How do I rationalize the denominator of a radical expression?

A: To rationalize the denominator of a radical expression, you need to multiply both the numerator and the denominator by the conjugate of the denominator.

Final Answer

The final simplified expression is:

(x+y)−116\boxed{(x+y)^{-\frac{11}{6}}}

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Introduction

In our previous article, we explored the concept of simplifying radical expressions and provided a step-by-step guide on how to simplify the given expression: ${ \frac{\sqrt[4]{(x+y)^2} \sqrt[3]{x+y}}{\sqrt{(x+y)^3}} }$. In this article, we will continue to provide more information and answer frequently asked questions related to simplifying radical expressions.

Q&A Session

Q: What is the main concept behind simplifying radical expressions?

A: The main concept behind simplifying radical expressions is to apply the rules of exponents and radicals to simplify each term individually and then combine the simplified terms.

Q: How do I simplify a radical expression involving multiple terms?

A: To simplify a radical expression involving multiple terms, you need to apply the rules of exponents and radicals to simplify each term individually and then combine the simplified terms.

Q: What is the importance of rationalizing the denominator?

A: Rationalizing the denominator is important because it helps to eliminate any radicals in the denominator, making the expression easier to work with.

Q: How do I rationalize the denominator of a radical expression?

A: To rationalize the denominator of a radical expression, you need to multiply both the numerator and the denominator by the conjugate of the denominator.

Q: What is the conjugate of a radical expression?

A: The conjugate of a radical expression is the expression with the opposite sign in the denominator.

Q: How do I find the conjugate of a radical expression?

A: To find the conjugate of a radical expression, you need to change the sign in the denominator.

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

A: A rational expression is an expression that involves fractions, while a radical expression is an expression that involves roots.

Q: How do I simplify a rational expression involving radicals?

A: To simplify a rational expression involving radicals, you need to apply the rules of exponents and radicals to simplify each term individually and then combine the simplified terms.

Q: What is the final simplified expression for the given problem?

A: The final simplified expression for the given problem is:

(x+y)−116\boxed{(x+y)^{-\frac{11}{6}}}

Tips and Tricks

Tip 1: Always start by simplifying the radical expression inside the parentheses.

Simplifying the radical expression inside the parentheses will make it easier to work with and simplify the entire expression.

Tip 2: Use the rules of exponents and radicals to simplify each term individually.

Applying the rules of exponents and radicals will help you simplify each term individually and then combine the simplified terms.

Tip 3: Rationalize the denominator to eliminate any radicals in the denominator.

Rationalizing the denominator will make the expression easier to work with and simplify.

Conclusion

In this article, we have provided more information and answered frequently asked questions related to simplifying radical expressions. We have also provided tips and tricks to help you simplify radical expressions efficiently.

Frequently Asked Questions

Q: What is the main concept behind simplifying radical expressions?

A: The main concept behind simplifying radical expressions is to apply the rules of exponents and radicals to simplify each term individually and then combine the simplified terms.

Q: How do I simplify a radical expression involving multiple terms?

A: To simplify a radical expression involving multiple terms, you need to apply the rules of exponents and radicals to simplify each term individually and then combine the simplified terms.

Q: What is the importance of rationalizing the denominator?

A: Rationalizing the denominator is important because it helps to eliminate any radicals in the denominator, making the expression easier to work with.

Q: How do I rationalize the denominator of a radical expression?

A: To rationalize the denominator of a radical expression, you need to multiply both the numerator and the denominator by the conjugate of the denominator.

Final Answer

The final simplified expression is:

(x+y)−116\boxed{(x+y)^{-\frac{11}{6}}}