Simplify The Following Expression Completely, Where X ≥ 0 X \geq 0 X ≥ 0 . X 5 X Y 4 + 405 X 3 Y 4 − 80 X 3 Y 4 X \sqrt{5 X Y^4} + \sqrt{405 X^3 Y^4} - \sqrt{80 X^3 Y^4} X 5 X Y 4 ​ + 405 X 3 Y 4 ​ − 80 X 3 Y 4 ​ Options:A. X Y 2 5 X + X Y 2 405 X − X Y 2 80 X X Y^2 \sqrt{5 X} + X Y^2 \sqrt{405 X} - X Y^2 \sqrt{80 X} X Y 2 5 X ​ + X Y 2 405 X ​ − X Y 2 80 X ​ B. $6 X Y^2 \sqrt{5

Introduction

Radical expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will delve into the world of radical expressions and explore a step-by-step guide on how to simplify them. We will focus on the given expression $x \sqrt{5 x y^4} + \sqrt{405 x^3 y^4} - \sqrt{80 x^3 y^4}$ and simplify it completely.

Understanding Radical Expressions

Before we dive into the simplification process, it's essential to understand the basics of radical expressions. A radical expression is a mathematical expression that contains a root or a radical sign. The most common radical sign is the square root sign (√). Radical expressions can be added, subtracted, multiplied, and divided just like regular expressions.

Simplifying the Given Expression

To simplify the given expression, we need to follow the order of operations (PEMDAS):

1. Parentheses: There are no parentheses in the given expression.
2. Exponents: There are no exponents in the given expression.
3. Multiplication and Division: We can start simplifying the expression by factoring out the common terms.
4. Addition and Subtraction: We will add and subtract the simplified terms.

Let's start simplifying the expression:

$x \sqrt{5 x y^4} + \sqrt{405 x^3 y^4} - \sqrt{80 x^3 y^4}$

Step 1: Factor out the Common Terms

We can factor out the common term $x^3 y^4$ from the second and third terms:

$x \sqrt{5 x y^4} + x^3 y^4 \sqrt{405} - x^3 y^4 \sqrt{80}$

Step 2: Simplify the Radicals

We can simplify the radicals by factoring out the perfect squares:

$x \sqrt{5 x y^4} + x^3 y^4 \sqrt{81 \cdot 5} - x^3 y^4 \sqrt{16 \cdot 5}$

$x \sqrt{5 x y^4} + 9 x^3 y^4 \sqrt{5} - 4 x^3 y^4 \sqrt{5}$

Step 3: Combine Like Terms

We can combine the like terms:

$x \sqrt{5 x y^4} + 5 x^3 y^4 \sqrt{5}$

Step 4: Factor out the Common Term

We can factor out the common term $5 x y^4$:

$5 x y^4 \sqrt{x} + 5 x^3 y^4 \sqrt{5}$

Step 5: Factor out the Common Term

We can factor out the common term $5 x y^4$:

$5 x y^4 (\sqrt{x} + x \sqrt{5})$

Step 6: Simplify the Expression

We can simplify the expression by factoring out the common term $x y^2$:

$5 x y^2 \sqrt{x} (1 + x \sqrt{5})$

Conclusion

In conclusion, the simplified expression is $5 x y^2 \sqrt{x} (1 + x \sqrt{5})$. This expression is the final answer to the given problem.

Comparison with the Options

Let's compare the simplified expression with the given options:

A. $x y^2 \sqrt{5 x} + x y^2 \sqrt{405 x} - x y^2 \sqrt{80 x}$

B. $6 x y^2 \sqrt{5 x} + 6 x y^2 \sqrt{405 x} - 6 x y^2 \sqrt{80 x}$

The simplified expression $5 x y^2 \sqrt{x} (1 + x \sqrt{5})$ is not equal to any of the given options. However, we can rewrite the expression as:

$5 x y^2 \sqrt{x} (1 + x \sqrt{5}) = 5 x y^2 \sqrt{x} + 5 x^2 y^2 \sqrt{5 x}$

This expression is similar to option A, but it's not exactly the same.

Discussion

The given problem is a classic example of simplifying radical expressions. The key to simplifying radical expressions is to follow the order of operations and factor out the common terms. In this case, we factored out the common term $5 x y^4$ and simplified the radicals. The final answer is $5 x y^2 \sqrt{x} (1 + x \sqrt{5})$.

Final Answer

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Introduction

Radical expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In our previous article, we explored a step-by-step guide on how to simplify the expression $x \sqrt{5 x y^4} + \sqrt{405 x^3 y^4} - \sqrt{80 x^3 y^4}$. In this article, we will answer some frequently asked questions about simplifying radical expressions.

Q&A

Q: What is the order of operations for simplifying radical expressions?

A: The order of operations for simplifying radical expressions is:

1. Parentheses: Evaluate expressions inside parentheses.
2. Exponents: Evaluate any exponential expressions.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

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

A: To simplify a radical expression with multiple terms, follow these steps:

1. Factor out the common terms: Factor out any common terms from the expression.
2. Simplify the radicals: Simplify each radical by factoring out perfect squares.
3. Combine like terms: Combine any like terms in the expression.

Q: What is the difference between a perfect square and a non-perfect square?

A: A perfect square is a number that can be expressed as the square of an integer, such as 4, 9, 16, etc. A non-perfect square is a number that cannot be expressed as the square of an integer, such as 2, 3, 5, etc.

Q: How do I simplify a radical expression with a variable in the radicand?

A: To simplify a radical expression with a variable in the radicand, follow these steps:

1. Factor out the variable: Factor out the variable from the radicand.
2. Simplify the radicals: Simplify each radical by factoring out perfect squares.
3. Combine like terms: Combine any like terms in the expression.

Q: What is the difference between a rational and irrational number?

A: A rational number is a number that can be expressed as the ratio of two integers, such as 3/4, 2/3, etc. An irrational number is a number that cannot be expressed as the ratio of two integers, such as the square root of 2, the square root of 3, etc.

Q: How do I simplify a radical expression with a rational exponent?

A: To simplify a radical expression with a rational exponent, follow these steps:

1. Simplify the radicals: Simplify each radical by factoring out perfect squares.
2. Apply the rational exponent: Apply the rational exponent to the simplified radicals.
3. Combine like terms: Combine any like terms in the expression.

Conclusion

Simplifying radical expressions is a crucial skill for any math enthusiast. By following the order of operations and factoring out common terms, you can simplify even the most complex radical expressions. Remember to simplify the radicals by factoring out perfect squares and combine like terms to get the final answer.

Final Tips

• Always follow the order of operations when simplifying radical expressions.
• Factor out common terms to simplify the expression.
• Simplify the radicals by factoring out perfect squares.
• Combine like terms to get the final answer.
• Practice, practice, practice! The more you practice simplifying radical expressions, the more comfortable you will become with the process.

Common Mistakes

• Not following the order of operations.
• Not factoring out common terms.
• Not simplifying the radicals by factoring out perfect squares.
• Not combining like terms.
• Not checking the final answer for errors.

Common Applications

• Simplifying radical expressions is a crucial skill for any math enthusiast.
• Radical expressions are used in a variety of real-world applications, such as physics, engineering, and computer science.
• Simplifying radical expressions can help you solve complex problems and make calculations easier.

Conclusion

Simplifying radical expressions is a fundamental concept in algebra, and it's essential to understand how to simplify them to solve complex problems. By following the order of operations and factoring out common terms, you can simplify even the most complex radical expressions. Remember to simplify the radicals by factoring out perfect squares and combine like terms to get the final answer. Practice, practice, practice! The more you practice simplifying radical expressions, the more comfortable you will become with the process.