Simplify The Expression: 7. − 44 \sqrt{-44} − 44 ​

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. When dealing with square roots, we often encounter expressions that can be simplified using various techniques. In this article, we will focus on simplifying the expression $\sqrt{-44}$.

Understanding Square Roots

Before we dive into simplifying the expression, let's quickly review what square roots are. 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 $4 \times 4 = 16$. Similarly, the square root of 25 is 5, because $5 \times 5 = 25$.

Simplifying Square Roots

To simplify a square root, we need to find the largest perfect square that divides the number inside the square root. A perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 is a perfect square because it can be expressed as $4 \times 4$, while 25 is a perfect square because it can be expressed as $5 \times 5$.

Simplifying $\sqrt{-44}$

Now, let's apply this technique to simplify the expression $\sqrt{-44}$. To do this, we need to find the largest perfect square that divides -44. We can start by factoring -44 into its prime factors: -44 = -2 × 2 × 11. We can see that -2 × 2 is a perfect square, because it can be expressed as $(-2) \times (-2) = 4$.

Using the Property of Square Roots

We can use the property of square roots that states $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ to simplify the expression. In this case, we can rewrite $\sqrt{-44}$ as $\sqrt{-2 \times 2 \times 11}$. Using the property, we can simplify this to $\sqrt{-2} \times \sqrt{2} \times \sqrt{11}$.

Simplifying $\sqrt{-2}$

Now, let's focus on simplifying $\sqrt{-2}$. We can rewrite $\sqrt{-2}$ as $\sqrt{-1} \times \sqrt{2}$, because $\sqrt{-1}$ is equal to $i$, the imaginary unit. Therefore, $\sqrt{-2} = i \times \sqrt{2}$.

Combining the Simplifications

Now that we have simplified $\sqrt{-2}$, we can combine the simplifications to get the final result. We have $\sqrt{-44} = \sqrt{-2} \times \sqrt{2} \times \sqrt{11} = i \times \sqrt{2} \times \sqrt{2} \times \sqrt{11}$. Using the property of square roots again, we can simplify this to $i \times 2 \times \sqrt{11} = 2i\sqrt{11}$.

Conclusion

In this article, we simplified the expression $\sqrt{-44}$ using various techniques. We started by factoring -44 into its prime factors and then used the property of square roots to simplify the expression. We also used the property of square roots to simplify $\sqrt{-2}$ and then combined the simplifications to get the final result. The final result is $2i\sqrt{11}$.

Frequently Asked Questions

• What is the square root of -44?
• How do you simplify a square root?
• What is the property of square roots?
• How do you simplify $\sqrt{-2}$?

Final Answer

The final answer is: $\boxed{2i\sqrt{11}}$

Introduction

In our previous article, we simplified the expression $\sqrt{-44}$ using various techniques. In this article, we will answer some frequently asked questions related to simplifying square roots and the expression $\sqrt{-44}$.

Q&A

Q: What is the square root of -44?

A: The square root of -44 is $2i\sqrt{11}$.

Q: How do you simplify a square root?

A: To simplify a square root, you need to find the largest perfect square that divides the number inside the square root. A perfect square is a number that can be expressed as the product of an integer with itself.

Q: What is the property of square roots?

A: The property of square roots states that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$. This means that you can simplify a square root by breaking it down into smaller square roots.

Q: How do you simplify $\sqrt{-2}$?

A: To simplify $\sqrt{-2}$, you can rewrite it as $\sqrt{-1} \times \sqrt{2}$, because $\sqrt{-1}$ is equal to $i$, the imaginary unit. Therefore, $\sqrt{-2} = i \times \sqrt{2}$.

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 is a number that can be expressed as the product of a real number and the imaginary unit $i$, such as $i$ or $2i$.

Q: Can you simplify $\sqrt{-44}$ using a different method?

A: Yes, you can simplify $\sqrt{-44}$ using a different method. One way to do this is to rewrite -44 as $-1 \times 44$. Then, you can simplify the square root of -1 to $i$, and the square root of 44 to $\sqrt{4 \times 11} = 2\sqrt{11}$. Therefore, $\sqrt{-44} = i \times 2\sqrt{11} = 2i\sqrt{11}$.

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

A: The imaginary unit $i$ is a fundamental concept in mathematics that allows us to extend the real number system to the complex number system. It is defined as the square root of -1, and it is used to represent numbers that cannot be expressed as real numbers.

Q: Can you provide more examples of simplifying square roots?

A: Yes, here are a few more examples:

• $\sqrt{-9} = \sqrt{-1} \times \sqrt{9} = i \times 3 = 3i$
• $\sqrt{-16} = \sqrt{-1} \times \sqrt{16} = i \times 4 = 4i$
• $\sqrt{-25} = \sqrt{-1} \times \sqrt{25} = i \times 5 = 5i$

Conclusion

In this article, we answered some frequently asked questions related to simplifying square roots and the expression $\sqrt{-44}$. We also provided some additional examples of simplifying square roots and discussed the significance of the imaginary unit $i$.

Frequently Asked Questions

• What is the square root of -44?
• How do you simplify a square root?
• What is the property of square roots?
• How do you simplify $\sqrt{-2}$?
• What is the difference between a real number and an imaginary number?
• Can you simplify $\sqrt{-44}$ using a different method?
• What is the significance of the imaginary unit $i$?
• Can you provide more examples of simplifying square roots?

Final Answer

The final answer is: $\boxed{2i\sqrt{11}}$