Simplify: $\sqrt{z^{36}}$Provide Your Answer Below:

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Introduction

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Radical expressions are an essential part of mathematics, and simplifying them is a crucial skill to master. In this article, we will focus on simplifying the radical expression $\sqrt{z^{36}}$. We will break down the process into manageable steps, making it easier to understand and apply.

Understanding the Problem

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The given expression is $\sqrt{z^{36}}$. To simplify this expression, we need to understand the properties of radicals and exponents. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we have the square root of $z^{36}$.

Breaking Down the Expression

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To simplify the expression, we can start by breaking it down into smaller parts. We can rewrite $z^{36}$ as $(z^{18})^2$. This is because $36$ is an even number, and we can express it as the product of two identical factors.

Applying the Property of Radicals

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Now that we have broken down the expression, we can apply the property of radicals. The square root of a number raised to an even power is equal to the number raised to half of that power. In this case, we have $\sqrt{(z^{18})^2}$, which simplifies to $z^{18}$.

Simplifying the Expression

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We have now simplified the expression to $z^{18}$. However, we can further simplify it by recognizing that $z^{18}$ can be expressed as $(z^9)^2$. This is because $18$ is an even number, and we can express it as the product of two identical factors.

Final Simplification

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We have now simplified the expression to $(z^9)^2$. However, we can further simplify it by recognizing that the square root of a number raised to an even power is equal to the number raised to half of that power. In this case, we have $\sqrt{(z^9)^2}$, which simplifies to $z^9$.

Conclusion

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In conclusion, we have simplified the radical expression $\sqrt{z^{36}}$ to $z^9$. We broke down the expression into smaller parts, applied the property of radicals, and simplified the expression step by step. This process demonstrates the importance of understanding the properties of radicals and exponents in simplifying complex expressions.

Frequently Asked Questions

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Q: What is the property of radicals?

A: The property of radicals states that the square root of a number raised to an even power is equal to the number raised to half of that power.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to break it down into smaller parts, apply the property of radicals, and simplify the expression step by step.

Q: What is the final simplified expression for $\sqrt{z^{36}}$?

A: The final simplified expression for $\sqrt{z^{36}}$ is $z^9$.

Additional Resources

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For more information on simplifying radical expressions, you can refer to the following resources:

• Mathway: A online math problem solver that can help you simplify radical expressions.
• Khan Academy: A free online learning platform that offers video lessons and practice exercises on simplifying radical expressions.
• Wolfram Alpha: A computational knowledge engine that can help you simplify radical expressions and provide step-by-step solutions.

Summary

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In this article, we simplified the radical expression $\sqrt{z^{36}}$ to $z^9$. We broke down the expression into smaller parts, applied the property of radicals, and simplified the expression step by step. This process demonstrates the importance of understanding the properties of radicals and exponents in simplifying complex expressions.

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Introduction

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In our previous article, we simplified the radical expression $\sqrt{z^{36}}$ to $z^9$. However, we understand that simplifying radical expressions can be a challenging task, and many of you may have questions about the process. In this article, we will address some of the most frequently asked questions about simplifying radical expressions.

Q&A

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Q: What is the property of radicals?

A: The property of radicals states that the square root of a number raised to an even power is equal to the number raised to half of that power. This means that if you have a number raised to an even power, you can simplify the square root of that number by taking the number to half of the power.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to follow these steps:

1. Break down the expression into smaller parts.
2. Identify the even powers in the expression.
3. Apply the property of radicals to simplify the expression.
4. Simplify the expression further by combining like terms.

Q: What is the difference between a radical and an exponent?

A: A radical is a symbol that represents the square root of a number, while an exponent is a number that represents the power to which a number is raised. For example, $\sqrt{z}$ is a radical, while $z^2$ is an exponent.

Q: Can I simplify a radical expression with an odd power?

A: No, you cannot simplify a radical expression with an odd power using the property of radicals. However, you can still simplify the expression by breaking it down into smaller parts and applying other mathematical operations.

Q: How do I handle negative numbers in radical expressions?

A: When dealing with negative numbers in radical expressions, you need to follow the rules of exponents. If the power is even, the negative sign can be removed, but if the power is odd, the negative sign remains.

Q: Can I simplify a radical expression with a variable?

A: Yes, you can simplify a radical expression with a variable. However, you need to follow the same steps as before: break down the expression into smaller parts, identify the even powers, and apply the property of radicals.

Q: What is the final simplified expression for $\sqrt{z^{36}}$?

A: The final simplified expression for $\sqrt{z^{36}}$ is $z^9$.

Q: How do I check my work when simplifying radical expressions?

A: To check your work, you can plug the simplified expression back into the original expression and see if it equals the original expression. You can also use a calculator or a computer program to check your work.

Additional Tips

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Tip 1: Break down the expression into smaller parts

When simplifying radical expressions, it's essential to break down the expression into smaller parts. This will make it easier to identify the even powers and apply the property of radicals.

Tip 2: Identify the even powers

Even powers are a crucial part of simplifying radical expressions. Make sure to identify the even powers in the expression and apply the property of radicals accordingly.

Tip 3: Apply the property of radicals

The property of radicals is a powerful tool for simplifying radical expressions. Make sure to apply it correctly to simplify the expression.

Tip 4: Simplify the expression further

Once you have simplified the expression using the property of radicals, you can further simplify it by combining like terms.

Conclusion

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Simplifying radical expressions can be a challenging task, but with practice and patience, you can master it. Remember to break down the expression into smaller parts, identify the even powers, and apply the property of radicals. With these tips and the Q&A guide, you'll be well on your way to simplifying radical expressions like a pro!

Frequently Asked Questions (FAQs)

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Q: What is the property of radicals?

A: The property of radicals states that the square root of a number raised to an even power is equal to the number raised to half of that power.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to break down the expression into smaller parts, identify the even powers, and apply the property of radicals.

Q: What is the difference between a radical and an exponent?

A: A radical is a symbol that represents the square root of a number, while an exponent is a number that represents the power to which a number is raised.

Additional Resources

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For more information on simplifying radical expressions, you can refer to the following resources:

• Mathway: A online math problem solver that can help you simplify radical expressions.
• Khan Academy: A free online learning platform that offers video lessons and practice exercises on simplifying radical expressions.
• Wolfram Alpha: A computational knowledge engine that can help you simplify radical expressions and provide step-by-step solutions.