What Is The Simplified Form Of The Following Expression? Assume $x \geq 0$ And $y \geq 0$.${ 2(\sqrt[4]{16x}) - 2(\sqrt[4]{2y}) + 3(\sqrt[4]{81x}) - 4(\sqrt[4]{32y}) }$A. $[ 5(\sqrt[4]{x}) - 4(\sqrt[4]{2y})

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Introduction

In this article, we will explore the simplified form of a given mathematical expression. The expression involves the use of fourth roots and various arithmetic operations. We will assume that x≥0x \geq 0 and y≥0y \geq 0 to simplify the expression.

Understanding the Expression

The given expression is:

2(16x4)−2(2y4)+3(81x4)−4(32y4){ 2(\sqrt[4]{16x}) - 2(\sqrt[4]{2y}) + 3(\sqrt[4]{81x}) - 4(\sqrt[4]{32y}) }

To simplify this expression, we need to understand the properties of fourth roots and how they can be manipulated.

Simplifying the Expression

Let's start by simplifying each term in the expression.

Simplifying the First Term

The first term is:

2(16x4){ 2(\sqrt[4]{16x}) }

We can simplify this term by using the property of fourth roots that states:

a44=a{ \sqrt[4]{a^4} = a }

In this case, we have:

16x4=(24)x4=2x4{ \sqrt[4]{16x} = \sqrt[4]{(2^4)x} = 2\sqrt[4]{x} }

So, the first term becomes:

2(2x4)=4x4{ 2(2\sqrt[4]{x}) = 4\sqrt[4]{x} }

Simplifying the Second Term

The second term is:

−2(2y4){ -2(\sqrt[4]{2y}) }

We can simplify this term by using the property of fourth roots that states:

a44=a{ \sqrt[4]{a^4} = a }

In this case, we have:

2y4=(21)y4=21/4y4{ \sqrt[4]{2y} = \sqrt[4]{(2^1)y} = 2^{1/4}\sqrt[4]{y} }

So, the second term becomes:

−2(21/4y4)=−25/4y4{ -2(2^{1/4}\sqrt[4]{y}) = -2^{5/4}\sqrt[4]{y} }

Simplifying the Third Term

The third term is:

3(81x4){ 3(\sqrt[4]{81x}) }

We can simplify this term by using the property of fourth roots that states:

a44=a{ \sqrt[4]{a^4} = a }

In this case, we have:

81x4=(34)x4=3x4{ \sqrt[4]{81x} = \sqrt[4]{(3^4)x} = 3\sqrt[4]{x} }

So, the third term becomes:

3(3x4)=9x4{ 3(3\sqrt[4]{x}) = 9\sqrt[4]{x} }

Simplifying the Fourth Term

The fourth term is:

−4(32y4){ -4(\sqrt[4]{32y}) }

We can simplify this term by using the property of fourth roots that states:

a44=a{ \sqrt[4]{a^4} = a }

In this case, we have:

32y4=(25)y4=25/4y4{ \sqrt[4]{32y} = \sqrt[4]{(2^5)y} = 2^{5/4}\sqrt[4]{y} }

So, the fourth term becomes:

−4(25/4y4)=−29/4y4{ -4(2^{5/4}\sqrt[4]{y}) = -2^{9/4}\sqrt[4]{y} }

Combining the Terms

Now that we have simplified each term, we can combine them to get the simplified form of the expression.

4x4−25/4y4+9x4−29/4y4{ 4\sqrt[4]{x} - 2^{5/4}\sqrt[4]{y} + 9\sqrt[4]{x} - 2^{9/4}\sqrt[4]{y} }

We can combine like terms to get:

(4+9)x4−(25/4+29/4)y4{ (4 + 9)\sqrt[4]{x} - (2^{5/4} + 2^{9/4})\sqrt[4]{y} }

Simplifying further, we get:

13x4−29/4y4{ 13\sqrt[4]{x} - 2^{9/4}\sqrt[4]{y} }

However, we can simplify the expression even further by factoring out a common term.

13x4−29/4y4=13x4−22⋅21/4y4{ 13\sqrt[4]{x} - 2^{9/4}\sqrt[4]{y} = 13\sqrt[4]{x} - 2^2\cdot2^{1/4}\sqrt[4]{y} }

=13x4−4⋅21/4y4{ = 13\sqrt[4]{x} - 4\cdot2^{1/4}\sqrt[4]{y} }

=13x4−42y4{ = 13\sqrt[4]{x} - 4\sqrt[4]{2y} }

Conclusion

In this article, we simplified the given expression by using the properties of fourth roots and various arithmetic operations. We assumed that x≥0x \geq 0 and y≥0y \geq 0 to simplify the expression. The simplified form of the expression is:

13x4−42y4{ 13\sqrt[4]{x} - 4\sqrt[4]{2y} }

This expression is the final answer to the problem.

Final Answer

Introduction

In our previous article, we simplified the given expression by using the properties of fourth roots and various arithmetic operations. We assumed that x≥0x \geq 0 and y≥0y \geq 0 to simplify the expression. In this article, we will answer some frequently asked questions related to the simplification of the given expression.

Q: What is the simplified form of the given expression?

A: The simplified form of the given expression is:

13x4−42y4{ 13\sqrt[4]{x} - 4\sqrt[4]{2y} }

Q: How did you simplify the expression?

A: We simplified the expression by using the properties of fourth roots and various arithmetic operations. We started by simplifying each term in the expression and then combined like terms to get the final simplified form.

Q: What is the assumption made in the simplification of the expression?

A: We assumed that x≥0x \geq 0 and y≥0y \geq 0 to simplify the expression.

Q: Can you explain the property of fourth roots used in the simplification?

A: Yes, the property of fourth roots used in the simplification is:

a44=a{ \sqrt[4]{a^4} = a }

This property allows us to simplify the expression by removing the fourth root from the terms.

Q: How did you combine like terms in the expression?

A: We combined like terms by adding or subtracting the coefficients of the terms with the same fourth root.

Q: Can you provide an example of how to simplify a similar expression?

A: Yes, here's an example of how to simplify a similar expression:

2(25x4)−3(8y4)+4(81x4)−5(32y4){ 2(\sqrt[4]{25x}) - 3(\sqrt[4]{8y}) + 4(\sqrt[4]{81x}) - 5(\sqrt[4]{32y}) }

To simplify this expression, we can follow the same steps as before:

  1. Simplify each term using the property of fourth roots.
  2. Combine like terms by adding or subtracting the coefficients of the terms with the same fourth root.

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

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

  • Not using the correct property of fourth roots.
  • Not combining like terms correctly.
  • Not assuming the correct conditions for the variables.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the given expression. We provided examples and explanations to help clarify the concepts and avoid common mistakes. We hope this article has been helpful in understanding the simplification of the given expression.

Final Answer

The final answer is 13x4−42y4\boxed{13\sqrt[4]{x} - 4\sqrt[4]{2y}}.