Simplify The Following Expression Completely, Where $x \geq 0$:$\[x \sqrt{5 X Y^4} + \sqrt{405 X^3 Y^4} - \sqrt{80 X^3 Y^4}\\]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

Introduction

In this article, we will delve into the world of algebra and simplify a given expression involving square roots. The expression is $x \sqrt{5 x y^4} + \sqrt{405 x^3 y^4} - \sqrt{80 x^3 y^4}$, where $x \geq 0$. Our goal is to simplify this expression completely and provide the final result in a clear and concise manner.

Step 1: Factor Out Common Terms

To simplify the given expression, we can start by factoring out common terms from each square root. We can rewrite the expression as follows:

${x \sqrt{5 x y^4} + \sqrt{405 x^3 y^4} - \sqrt{80 x^3 y^4} = x \sqrt{5 x y^4} + \sqrt{81 x^3 y^4 \cdot 5} - \sqrt{16 x^3 y^4 \cdot 5}}$

Step 2: Simplify the Square Roots

Now, we can simplify each square root by factoring out the perfect squares. We can rewrite the expression as follows:

${x \sqrt{5 x y^4} + \sqrt{81 x^3 y^4 \cdot 5} - \sqrt{16 x^3 y^4 \cdot 5} = x \sqrt{5 x y^4} + 9 x \sqrt{x^2 y^4 \cdot 5} - 4 x \sqrt{x^2 y^4 \cdot 5}}$

Step 3: Combine Like Terms

Next, we can combine like terms by factoring out the common term $x \sqrt{x^2 y^4 \cdot 5}$. We can rewrite the expression as follows:

${x \sqrt{5 x y^4} + 9 x \sqrt{x^2 y^4 \cdot 5} - 4 x \sqrt{x^2 y^4 \cdot 5} = x \sqrt{5 x y^4} + (9 x - 4 x) \sqrt{x^2 y^4 \cdot 5}}$

Step 4: Simplify the Expression

Finally, we can simplify the expression by evaluating the expression inside the square root. We can rewrite the expression as follows:

${x \sqrt{5 x y^4} + (9 x - 4 x) \sqrt{x^2 y^4 \cdot 5} = x \sqrt{5 x y^4} + 5 x \sqrt{x^2 y^4 \cdot 5}}$

Step 5: Factor Out Common Terms

We can factor out the common term $x \sqrt{x^2 y^4 \cdot 5}$ from the expression. We can rewrite the expression as follows:

${x \sqrt{5 x y^4} + 5 x \sqrt{x^2 y^4 \cdot 5} = x \sqrt{5 x y^4} + 5 x \sqrt{5 x^2 y^4}}$

Step 6: Simplify the Expression

We can simplify the expression by evaluating the expression inside the square root. We can rewrite the expression as follows:

${x \sqrt{5 x y^4} + 5 x \sqrt{5 x^2 y^4} = x y^2 \sqrt{5 x} + 5 x y^2 \sqrt{5 x}}$

Step 7: Combine Like Terms

Finally, we can combine like terms by factoring out the common term $x y^2 \sqrt{5 x}$. We can rewrite the expression as follows:

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

Step 8: Simplify the Expression

We can simplify the expression by evaluating the expression inside the parentheses. We can rewrite the expression as follows:

${(x y^2 \sqrt{5 x} + 5 x y^2 \sqrt{5 x}) = 6 x y^2 \sqrt{5 x}}$

Conclusion

In this article, we simplified the given expression involving square roots. We started by factoring out common terms, then simplified each square root by factoring out perfect squares. We combined like terms and finally simplified the expression by evaluating the expression inside the parentheses. The final result is $6 x y^2 \sqrt{5 x}$.

Answer

The correct answer is:

${6 x y^2 \sqrt{5 x}}$

Introduction

In our previous article, we simplified the given expression involving square roots. We started by factoring out common terms, then simplified each square root by factoring out perfect squares. We combined like terms and finally simplified the expression by evaluating the expression inside the parentheses. In this article, we will provide a Q&A approach to help you understand the simplification process.

Q: What is the given expression?

A: The given expression is $x \sqrt{5 x y^4} + \sqrt{405 x^3 y^4} - \sqrt{80 x^3 y^4}$, where $x \geq 0$.

Q: How do we simplify the given expression?

A: We can simplify the given expression by factoring out common terms, then simplifying each square root by factoring out perfect squares. We can combine like terms and finally simplify the expression by evaluating the expression inside the parentheses.

Q: What is the first step in simplifying the given expression?

A: The first step in simplifying the given expression is to factor out common terms. We can rewrite the expression as follows:

${x \sqrt{5 x y^4} + \sqrt{405 x^3 y^4} - \sqrt{80 x^3 y^4} = x \sqrt{5 x y^4} + \sqrt{81 x^3 y^4 \cdot 5} - \sqrt{16 x^3 y^4 \cdot 5}}$

Q: How do we simplify the square roots?

A: We can simplify each square root by factoring out the perfect squares. We can rewrite the expression as follows:

${x \sqrt{5 x y^4} + \sqrt{81 x^3 y^4 \cdot 5} - \sqrt{16 x^3 y^4 \cdot 5} = x \sqrt{5 x y^4} + 9 x \sqrt{x^2 y^4 \cdot 5} - 4 x \sqrt{x^2 y^4 \cdot 5}}$

Q: How do we combine like terms?

A: We can combine like terms by factoring out the common term $x \sqrt{x^2 y^4 \cdot 5}$. We can rewrite the expression as follows:

${x \sqrt{5 x y^4} + 9 x \sqrt{x^2 y^4 \cdot 5} - 4 x \sqrt{x^2 y^4 \cdot 5} = x \sqrt{5 x y^4} + (9 x - 4 x) \sqrt{x^2 y^4 \cdot 5}}$

Q: How do we simplify the expression?

A: We can simplify the expression by evaluating the expression inside the parentheses. We can rewrite the expression as follows:

${x \sqrt{5 x y^4} + (9 x - 4 x) \sqrt{x^2 y^4 \cdot 5} = x \sqrt{5 x y^4} + 5 x \sqrt{x^2 y^4 \cdot 5}}$

Q: How do we factor out common terms?

A: We can factor out the common term $x \sqrt{x^2 y^4 \cdot 5}$ from the expression. We can rewrite the expression as follows:

${x \sqrt{5 x y^4} + 5 x \sqrt{x^2 y^4 \cdot 5} = x \sqrt{5 x y^4} + 5 x \sqrt{5 x^2 y^4}}$

Q: How do we simplify the expression?

A: We can simplify the expression by evaluating the expression inside the square root. We can rewrite the expression as follows:

${x \sqrt{5 x y^4} + 5 x \sqrt{5 x^2 y^4} = x y^2 \sqrt{5 x} + 5 x y^2 \sqrt{5 x}}$

Q: How do we combine like terms?

A: We can combine like terms by factoring out the common term $x y^2 \sqrt{5 x}$. We can rewrite the expression as follows:

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

Q: How do we simplify the expression?

A: We can simplify the expression by evaluating the expression inside the parentheses. We can rewrite the expression as follows:

${(x y^2 \sqrt{5 x} + 5 x y^2 \sqrt{5 x}) = 6 x y^2 \sqrt{5 x}}$

Conclusion

In this article, we provided a Q&A approach to help you understand the simplification process of the given expression involving square roots. We started by factoring out common terms, then simplified each square root by factoring out perfect squares. We combined like terms and finally simplified the expression by evaluating the expression inside the parentheses. The final result is $6 x y^2 \sqrt{5 x}$.

Answer

The correct answer is:

${6 x y^2 \sqrt{5 x}}$

This is option B.