Simplify: X 2 Y 6 \sqrt{x^2 Y^6} X 2 Y 6 ​ A. ∣ X ∣ Y 2 |x| Y^2 ∣ X ∣ Y 2 B. ∣ X Y 3 ∣ \left|x Y^3\right| ​ X Y 3 ​ C. ∣ X ∣ Y 4 |x| Y^4 ∣ X ∣ Y 4 D. ∣ X Y 6 ∣ \left|x Y^6\right| ​ X Y 6 ​

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Understanding the Problem

When dealing with square roots, it's essential to remember that the square root of a number is the value that, when multiplied by itself, gives the original number. In mathematical terms, $\sqrt{a} = b$ implies that $b^2 = a$. This concept is crucial in simplifying expressions involving square roots.

Breaking Down the Expression

The given expression is $\sqrt{x^2 y^6}$. To simplify this, we need to understand the properties of exponents and square roots. The expression can be broken down into two separate terms: $x^2$ and $y^6$. We can then apply the property of square roots that states $\sqrt{a^2} = |a|$.

Applying the Property of Square Roots

Using the property mentioned above, we can rewrite the expression as $\sqrt{x^2} \cdot \sqrt{y^6}$. This simplifies to $|x| \cdot \sqrt{y^6}$.

Simplifying the Square Root of $y^6$

Now, we need to simplify the square root of $y^6$. Since the square root of a number is the value that, when multiplied by itself, gives the original number, we can rewrite $\sqrt{y^6}$ as $y^3$. This is because $y^3 \cdot y^3 = y^6$.

Combining the Terms

Now that we have simplified the square root of $y^6$, we can combine the terms to get the final simplified expression. We have $|x| \cdot y^3$, which can be rewritten as $\left|x y^3\right|$.

Conclusion

In conclusion, the simplified expression for $\sqrt{x^2 y^6}$ is $\left|x y^3\right|$. This is the correct answer among the given options.

Comparison with Options

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

• A. $|x| y^2$: This option is incorrect because the simplified expression has $y^3$, not $y^2$.
• B. $\left|x y^3\right|$ : This option is correct because it matches our simplified expression.
• C. $|x| y^4$: This option is incorrect because the simplified expression has $y^3$, not $y^4$.
• D. $\left|x y^6\right|$ : This option is incorrect because the simplified expression has $y^3$, not $y^6$.

Final Answer

The final answer is $\boxed{\left|x y^3\right|}$.

Understanding the Concept

The concept of simplifying expressions involving square roots is crucial in mathematics. It requires a deep understanding of the properties of exponents and square roots. By applying these properties, we can simplify complex expressions and arrive at the correct solution.

Real-World Applications

The concept of simplifying expressions involving square roots has numerous real-world applications. For example, in physics, the square root of a number is used to calculate the velocity of an object. In engineering, the square root of a number is used to calculate the stress on a material.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions involving square roots:

• Always start by breaking down the expression into separate terms.
• Apply the property of square roots that states $\sqrt{a^2} = |a|$.
• Simplify the square root of each term separately.
• Combine the simplified terms to get the final expression.

Common Mistakes

Here are some common mistakes to avoid when simplifying expressions involving square roots:

• Not breaking down the expression into separate terms.
• Not applying the property of square roots correctly.
• Not simplifying the square root of each term separately.
• Not combining the simplified terms correctly.

Conclusion

In conclusion, simplifying expressions involving square roots requires a deep understanding of the properties of exponents and square roots. By applying these properties and following the tips and tricks mentioned above, you can simplify complex expressions and arrive at the correct solution.

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Frequently Asked Questions

Q: What is the property of square roots that we need to apply to simplify the expression $\sqrt{x^2 y^6}$?

A: The property of square roots that we need to apply is $\sqrt{a^2} = |a|$. This property states that the square root of a number squared is equal to the absolute value of the number.

Q: How do we simplify the square root of $y^6$?

A: We can simplify the square root of $y^6$ by rewriting it as $y^3$. This is because $y^3 \cdot y^3 = y^6$.

Q: What is the final simplified expression for $\sqrt{x^2 y^6}$?

A: The final simplified expression for $\sqrt{x^2 y^6}$ is $\left|x y^3\right|$.

Q: Why is the option $|x| y^2$ incorrect?

A: The option $|x| y^2$ is incorrect because the simplified expression has $y^3$, not $y^2$.

Q: Why is the option $\left|x y^6\right|$ incorrect?

A: The option $\left|x y^6\right|$ is incorrect because the simplified expression has $y^3$, not $y^6$.

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:

• Not breaking down the expression into separate terms.
• Not applying the property of square roots correctly.
• Not simplifying the square root of each term separately.
• Not combining the simplified terms correctly.

Q: How can we apply the property of square roots to simplify complex expressions?

A: We can apply the property of square roots to simplify complex expressions by breaking down the expression into separate terms, applying the property of square roots to each term, and then combining the simplified terms.

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:

• Calculating the velocity of an object in physics.
• Calculating the stress on a material in engineering.

Q: What are some tips and tricks to help us simplify expressions involving square roots?

A: Some tips and tricks to help us simplify expressions involving square roots include:

• Always start by breaking down the expression into separate terms.
• Apply the property of square roots that states $\sqrt{a^2} = |a|$.
• Simplify the square root of each term separately.
• Combine the simplified terms to get the final expression.

Conclusion

In conclusion, simplifying expressions involving square roots requires a deep understanding of the properties of exponents and square roots. By applying these properties and following the tips and tricks mentioned above, you can simplify complex expressions and arrive at the correct solution.

Additional Resources

• For more information on simplifying expressions involving square roots, check out the following resources:

• Khan Academy: Simplifying Square Roots
• Mathway: Simplifying Square Roots
• Wolfram Alpha: Simplifying Square Roots

Final Answer

The final answer is $\boxed{\left|x y^3\right|}$.