Simplify The Expression 9 X 4 Y 6 \sqrt{9 X^4 Y^6} 9 X 4 Y 6 ​ .A. 3 3 3 B. 3 X Y 2 3 X Y^2 3 X Y 2 C. 3 X 2 Y 6 3 X^2 \sqrt{y^6} 3 X 2 Y 6 ​ D. 3 X 2 Y 3 3 X^2 Y^3 3 X 2 Y 3

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Understanding the Problem

When simplifying an expression involving square roots, it's essential to identify perfect squares within the radicand. This allows us to simplify the expression by taking the square root of the perfect squares and leaving the remaining factors inside the square root. In this problem, we're given the expression $\sqrt{9 x^4 y^6}$ and need to simplify it.

Breaking Down the Expression

To simplify the expression, we need to break it down into its prime factors. The expression can be written as $\sqrt{9 x^4 y^6} = \sqrt{3^2 \cdot x^4 \cdot y^6}$. Now, we can identify the perfect squares within the radicand.

Identifying Perfect Squares

The perfect squares within the radicand are $3^2$ and $x^4$. We can take the square root of these perfect squares, which gives us $3$ and $x^2$, respectively.

Simplifying the Expression

Now that we've identified the perfect squares, we can simplify the expression by taking the square root of these perfect squares and leaving the remaining factors inside the square root. This gives us $\sqrt{3^2 \cdot x^4 \cdot y^6} = 3 x^2 \sqrt{y^6}$.

Further Simplification

We can further simplify the expression by taking the square root of $y^6$. Since $y^6$ is a perfect square, we can take the square root of it, which gives us $y^3$.

Final Simplification

Now that we've taken the square root of $y^6$, we can simplify the expression further. This gives us $3 x^2 \sqrt{y^6} = 3 x^2 y^3$.

Conclusion

In conclusion, the simplified expression is $3 x^2 y^3$. This is the final answer to the problem.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Break down the expression into its prime factors: $\sqrt{9 x^4 y^6} = \sqrt{3^2 \cdot x^4 \cdot y^6}$.
2. Identify the perfect squares within the radicand: $3^2$ and $x^4$.
3. Take the square root of the perfect squares: $3$ and $x^2$.
4. Simplify the expression by taking the square root of the perfect squares and leaving the remaining factors inside the square root: $\sqrt{3^2 \cdot x^4 \cdot y^6} = 3 x^2 \sqrt{y^6}$.
5. Take the square root of $y^6$: $y^3$.
6. Simplify the expression further: $3 x^2 \sqrt{y^6} = 3 x^2 y^3$.

Common Mistakes to Avoid

When simplifying expressions involving square roots, it's essential to identify perfect squares within the radicand. Failing to do so can lead to incorrect simplifications. Additionally, it's crucial to take the square root of the perfect squares and leave the remaining factors inside the square root.

Real-World Applications

Simplifying expressions involving square roots has numerous real-world applications. For example, in physics, the square root of a quantity is often used to represent the magnitude of a vector. In engineering, the square root of a quantity is used to represent the magnitude of a force or a stress.

Final Answer

The final answer to the problem is $\boxed{3 x^2 y^3}$.

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Understanding the Problem

When simplifying an expression involving square roots, it's essential to identify perfect squares within the radicand. This allows us to simplify the expression by taking the square root of the perfect squares and leaving the remaining factors inside the square root. In this problem, we're given the expression $\sqrt{9 x^4 y^6}$ and need to simplify it.

Q&A

Q: What is the first step in simplifying the expression $\sqrt{9 x^4 y^6}$?

A: The first step in simplifying the expression is to break it down into its prime factors. This gives us $\sqrt{9 x^4 y^6} = \sqrt{3^2 \cdot x^4 \cdot y^6}$.

Q: How do we identify perfect squares within the radicand?

A: We identify perfect squares within the radicand by looking for factors that are perfect squares. In this case, we have $3^2$ and $x^4$ as perfect squares.

Q: What is the next step in simplifying the expression?

A: The next step in simplifying the expression is to take the square root of the perfect squares. This gives us $3$ and $x^2$.

Q: How do we simplify the expression further?

A: We simplify the expression further by taking the square root of $y^6$. This gives us $y^3$.

Q: What is the final simplified expression?

A: The final simplified expression is $3 x^2 y^3$.

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

A: Some common mistakes to avoid when simplifying expressions involving square roots include failing to identify perfect squares within the radicand and not taking the square root of the perfect squares.

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

A: Some real-world applications of simplifying expressions involving square roots include representing the magnitude of a vector in physics and representing the magnitude of a force or a stress in engineering.

Additional Tips and Tricks

Tip 1: Identify Perfect Squares

When simplifying expressions involving square roots, it's essential to identify perfect squares within the radicand. This allows us to simplify the expression by taking the square root of the perfect squares and leaving the remaining factors inside the square root.

Tip 2: Take the Square Root of Perfect Squares

Once we've identified the perfect squares, we need to take the square root of them. This gives us the simplified expression.

Tip 3: Simplify Further

After taking the square root of the perfect squares, we may need to simplify the expression further by taking the square root of any remaining factors.

Conclusion

In conclusion, simplifying expressions involving square roots requires identifying perfect squares within the radicand and taking the square root of them. By following these steps and avoiding common mistakes, we can simplify expressions involving square roots and apply them to real-world problems.

Final Answer

The final answer to the problem is $\boxed{3 x^2 y^3}$.