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

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Understanding the Expression

The given expression involves the fourth roots of various numbers, along with some coefficients. To simplify this expression, we need to apply the properties of exponents and roots. The expression is as follows:

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

Breaking Down the Expression

Let's break down the expression into smaller parts to simplify it. We can start by simplifying the terms inside the parentheses.

Simplifying the Terms Inside the Parentheses

We can rewrite the terms inside the parentheses using the properties of exponents and roots.

  • 16x4=24x4=2x4\sqrt[4]{16 x} = \sqrt[4]{2^4 x} = 2\sqrt[4]{x}
  • 2y4=21y4=24y4\sqrt[4]{2 y} = \sqrt[4]{2^1 y} = \sqrt[4]{2} \sqrt[4]{y}
  • 81x4=34x4=3x4\sqrt[4]{81 x} = \sqrt[4]{3^4 x} = 3\sqrt[4]{x}
  • 32y4=25y4=223y4=28y4=42y4\sqrt[4]{32 y} = \sqrt[4]{2^5 y} = 2\sqrt[4]{2^3 y} = 2\sqrt[4]{8 y} = 4\sqrt[4]{2 y}

Substituting the Simplified Terms

Now that we have simplified the terms inside the parentheses, we can substitute them back into the original expression.

2(2x4)−2(24y4)+3(3x4)−4(42y4)2(2\sqrt[4]{x}) - 2(\sqrt[4]{2} \sqrt[4]{y}) + 3(3\sqrt[4]{x}) - 4(4\sqrt[4]{2 y})

Simplifying the Expression

We can simplify the expression further by applying the properties of exponents and roots.

4x4−224y4+9x4−162y44\sqrt[4]{x} - 2\sqrt[4]{2} \sqrt[4]{y} + 9\sqrt[4]{x} - 16\sqrt[4]{2 y}

Combining Like Terms

We can combine like terms to simplify the expression further.

(4+9)x4−(2+16)2y4(4 + 9)\sqrt[4]{x} - (2 + 16)\sqrt[4]{2 y}

Simplifying the Coefficients

We can simplify the coefficients by combining them.

13x4−182y413\sqrt[4]{x} - 18\sqrt[4]{2 y}

Conclusion

The simplified form of the given expression is 13x4−182y413\sqrt[4]{x} - 18\sqrt[4]{2 y}.

Final Answer

The final answer is 13x4−182y4\boxed{13\sqrt[4]{x} - 18\sqrt[4]{2 y}}.

Discussion

The given expression involves the fourth roots of various numbers, along with some coefficients. To simplify this expression, we need to apply the properties of exponents and roots. We can break down the expression into smaller parts to simplify it. By simplifying the terms inside the parentheses and combining like terms, we can arrive at the simplified form of the expression.

Key Takeaways

  • The properties of exponents and roots can be used to simplify expressions involving roots.
  • Breaking down the expression into smaller parts can make it easier to simplify.
  • Combining like terms can simplify the expression further.

Related Topics

  • Properties of exponents and roots
  • Simplifying expressions involving roots
  • Combining like terms

Further Reading

Q: What are the properties of exponents and roots that can be used to simplify expressions involving roots?

A: The properties of exponents and roots that can be used to simplify expressions involving roots include:

  • The product of two numbers raised to a power is equal to the product of the two numbers raised to that power.
  • The quotient of two numbers raised to a power is equal to the quotient of the two numbers raised to that power.
  • The power of a product is equal to the product of the powers.
  • The power of a quotient is equal to the quotient of the powers.

Q: How can I simplify an expression involving roots by breaking it down into smaller parts?

A: To simplify an expression involving roots by breaking it down into smaller parts, follow these steps:

  1. Identify the terms inside the parentheses.
  2. Simplify each term inside the parentheses using the properties of exponents and roots.
  3. Combine like terms to simplify the expression further.

Q: What is the difference between a like term and a unlike term?

A: A like term is a term that has the same variable raised to the same power. A unlike term is a term that has a different variable raised to a different power.

Q: How can I combine like terms to simplify an expression involving roots?

A: To combine like terms to simplify an expression involving roots, follow these steps:

  1. Identify the like terms in the expression.
  2. Add or subtract the coefficients of the like terms.
  3. Keep the variable and the power the same.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is 13x4−182y4\boxed{13\sqrt[4]{x} - 18\sqrt[4]{2 y}}.

Q: What are some related topics to the simplification of expressions involving roots?

A: Some related topics to the simplification of expressions involving roots include:

  • Properties of exponents and roots
  • Simplifying expressions involving roots
  • Combining like terms

Q: Where can I find further reading on the simplification of expressions involving roots?

A: You can find further reading on the simplification of expressions involving roots at the following resources:

Q: What are some key takeaways from the simplification of expressions involving roots?

A: Some key takeaways from the simplification of expressions involving roots include:

  • The properties of exponents and roots can be used to simplify expressions involving roots.
  • Breaking down the expression into smaller parts can make it easier to simplify.
  • Combining like terms can simplify the expression further.

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

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

  • Not using the properties of exponents and roots to simplify the expression.
  • Not breaking down the expression into smaller parts.
  • Not combining like terms.

Q: How can I practice simplifying expressions involving roots?

A: You can practice simplifying expressions involving roots by:

  • Working through examples and exercises.
  • Using online resources and tools.
  • Seeking help from a teacher or tutor.

Q: What are some real-world applications of simplifying expressions involving roots?

A: Some real-world applications of simplifying expressions involving roots include:

  • Calculating the area and perimeter of shapes.
  • Determining the volume of objects.
  • Solving problems in physics and engineering.