Simplify \[$\sqrt{-72}\$\].

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Introduction

Simplifying square roots can be a challenging task, especially when dealing with negative numbers. In this article, we will explore the process of simplifying $\sqrt{-72}$, which involves understanding the properties of square roots and the concept of imaginary numbers.

Understanding Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. However, when dealing with negative numbers, the concept of square roots becomes more complex.

Simplifying Square Roots of Negative Numbers

To simplify $\sqrt{-72}$, we need to understand that the square root of a negative number is an imaginary number. An imaginary number is a complex number that, when squared, gives a negative result. In this case, we can rewrite $\sqrt{-72}$ as $\sqrt{-1} \cdot \sqrt{72}$.

Properties of Imaginary Numbers

The square root of -1 is denoted by $i$, where $i^2 = -1$. This means that $i$ is an imaginary unit, and any power of $i$ can be simplified using the following properties:

• $i^2 = -1$
• $i^3 = -i$
• $i^4 = 1$

Simplifying $\sqrt{72}$

To simplify $\sqrt{72}$, we need to find the largest perfect square that divides 72. In this case, the largest perfect square that divides 72 is 36. We can rewrite $\sqrt{72}$ as $\sqrt{36} \cdot \sqrt{2}$.

Simplifying $\sqrt{36}$

The square root of 36 is 6, because 6 multiplied by 6 equals 36. Therefore, we can simplify $\sqrt{36}$ as 6.

Combining the Simplifications

Now that we have simplified $\sqrt{72}$ as $\sqrt{36} \cdot \sqrt{2}$, we can substitute the value of $\sqrt{36}$ as 6. This gives us $\sqrt{72} = 6\sqrt{2}$.

Simplifying $\sqrt{-72}$

Now that we have simplified $\sqrt{72}$ as $6\sqrt{2}$, we can substitute this value into the original expression $\sqrt{-72} = \sqrt{-1} \cdot \sqrt{72}$. This gives us $\sqrt{-72} = i \cdot 6\sqrt{2}$.

Final Simplification

To simplify $i \cdot 6\sqrt{2}$, we can multiply the two values together. This gives us $i \cdot 6\sqrt{2} = 6i\sqrt{2}$.

Conclusion

In conclusion, simplifying $\sqrt{-72}$ involves understanding the properties of square roots and the concept of imaginary numbers. By rewriting $\sqrt{-72}$ as $\sqrt{-1} \cdot \sqrt{72}$, we can simplify the expression using the properties of imaginary numbers and the concept of perfect squares. The final simplification of $\sqrt{-72}$ is $6i\sqrt{2}$.

Additional Resources

For more information on simplifying square roots and imaginary numbers, please refer to the following resources:

• Wikipedia: Square Root
• Wikipedia: Imaginary Unit
• Khan Academy: Simplifying Square Roots

Frequently Asked Questions

• Q: What is the square root of -1? A: The square root of -1 is denoted by $i$, where $i^2 = -1$.
• Q: How do I simplify $\sqrt{-72}$? A: To simplify $\sqrt{-72}$, you can rewrite it as $\sqrt{-1} \cdot \sqrt{72}$ and then simplify the expression using the properties of imaginary numbers and the concept of perfect squares.
• Q: What is the final simplification of $\sqrt{-72}$? A: The final simplification of $\sqrt{-72}$ is $6i\sqrt{2}$.
  

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Introduction

In our previous article, we explored the process of simplifying $\sqrt{-72}$, which involves understanding the properties of square roots and the concept of imaginary numbers. In this article, we will answer some frequently asked questions related to simplifying square roots and imaginary numbers.

Q&A

Q: What is the square root of -1?

A: The square root of -1 is denoted by $i$, where $i^2 = -1$. This means that $i$ is an imaginary unit, and any power of $i$ can be simplified using the following properties:

• $i^2 = -1$
• $i^3 = -i$
• $i^4 = 1$

Q: How do I simplify $\sqrt{-72}$?

A: To simplify $\sqrt{-72}$, you can rewrite it as $\sqrt{-1} \cdot \sqrt{72}$ and then simplify the expression using the properties of imaginary numbers and the concept of perfect squares.

Q: What is the largest perfect square that divides 72?

A: The largest perfect square that divides 72 is 36. This means that $\sqrt{72}$ can be rewritten as $\sqrt{36} \cdot \sqrt{2}$.

Q: How do I simplify $\sqrt{36}$?

A: The square root of 36 is 6, because 6 multiplied by 6 equals 36. Therefore, we can simplify $\sqrt{36}$ as 6.

Q: What is the final simplification of $\sqrt{-72}$?

A: The final simplification of $\sqrt{-72}$ is $6i\sqrt{2}$.

Q: Can I simplify $\sqrt{-72}$ using a different method?

A: Yes, you can simplify $\sqrt{-72}$ using a different method. One way to do this is to rewrite $\sqrt{-72}$ as $\sqrt{-1} \cdot \sqrt{72}$ and then simplify the expression using the properties of imaginary numbers and the concept of perfect squares.

Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be expressed as a decimal or a fraction, such as 3 or 1/2. An imaginary number, on the other hand, is a complex number that, when squared, gives a negative result. In this case, $i$ is an imaginary unit, and any power of $i$ can be simplified using the following properties:

• $i^2 = -1$
• $i^3 = -i$
• $i^4 = 1$

Q: Can I use a calculator to simplify $\sqrt{-72}$?

A: Yes, you can use a calculator to simplify $\sqrt{-72}$. However, keep in mind that calculators may not always give you the exact answer, especially when dealing with complex numbers.

Q: What is the significance of the imaginary unit $i$?

A: The imaginary unit $i$ is a fundamental concept in mathematics, and it plays a crucial role in many areas of mathematics, including algebra, geometry, and calculus. $i$ is used to extend the real number system to the complex number system, which allows us to solve equations that cannot be solved using only real numbers.

Conclusion

In conclusion, simplifying $\sqrt{-72}$ involves understanding the properties of square roots and the concept of imaginary numbers. By rewriting $\sqrt{-72}$ as $\sqrt{-1} \cdot \sqrt{72}$ and then simplifying the expression using the properties of imaginary numbers and the concept of perfect squares, we can arrive at the final simplification of $6i\sqrt{2}$. We hope that this article has provided you with a better understanding of how to simplify $\sqrt{-72}$ and has answered some of the frequently asked questions related to this topic.

Additional Resources

For more information on simplifying square roots and imaginary numbers, please refer to the following resources:

• Wikipedia: Square Root
• Wikipedia: Imaginary Unit
• Khan Academy: Simplifying Square Roots

Frequently Asked Questions

• Q: What is the square root of -1? A: The square root of -1 is denoted by $i$, where $i^2 = -1$.
• Q: How do I simplify $\sqrt{-72}$? A: To simplify $\sqrt{-72}$, you can rewrite it as $\sqrt{-1} \cdot \sqrt{72}$ and then simplify the expression using the properties of imaginary numbers and the concept of perfect squares.
• Q: What is the final simplification of $\sqrt{-72}$? A: The final simplification of $\sqrt{-72}$ is $6i\sqrt{2}$.