Write − 147 \sqrt{-147} − 147 ​ In Simplest Radical Form.

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Introduction

When dealing with square roots of negative numbers, we often encounter the concept of imaginary numbers. In this case, we are tasked with simplifying the expression −147\sqrt{-147} into its simplest radical form. To achieve this, we will first break down the number under the square root sign into its prime factors.

Breaking Down the Number

The number −147-147 can be expressed as the product of its prime factors. To do this, we start by dividing the number by the smallest prime number, which is 22. However, since 147147 is an odd number, we move on to the next prime number, which is 33. We can divide 147147 by 33 to get 4949. Since 4949 is a perfect square, we can further simplify it by taking the square root of 4949, which is 77. Therefore, we can express −147-147 as −3⋅72-3 \cdot 7^2.

Simplifying the Square Root

Now that we have broken down the number −147-147 into its prime factors, we can simplify the square root expression. We can rewrite −147\sqrt{-147} as −3⋅72\sqrt{-3 \cdot 7^2}. Using the property of square roots that allows us to separate the square root of a product into the product of the square roots, we can rewrite this expression as −3⋅72\sqrt{-3} \cdot \sqrt{7^2}.

Simplifying the Square Root of a Negative Number

When dealing with the square root of a negative number, we often encounter the concept of imaginary numbers. In this case, we can express −3\sqrt{-3} as i3i\sqrt{3}, where ii is the imaginary unit. The imaginary unit ii is defined as the square root of −1-1, and it is used to extend the real number system to the complex number system.

Simplifying the Expression

Now that we have simplified the square root of the negative number, we can simplify the entire expression. We can rewrite −147\sqrt{-147} as i3⋅72i\sqrt{3} \cdot \sqrt{7^2}. Using the property of square roots that allows us to separate the square root of a product into the product of the square roots, we can rewrite this expression as i3⋅7i\sqrt{3} \cdot 7.

Conclusion

In conclusion, we have successfully simplified the expression −147\sqrt{-147} into its simplest radical form. We first broke down the number under the square root sign into its prime factors, then simplified the square root expression using the property of square roots. Finally, we simplified the square root of the negative number using the concept of imaginary numbers. The simplified expression is i3⋅7i\sqrt{3} \cdot 7.

Final Answer

The final answer is i3⋅7\boxed{i\sqrt{3} \cdot 7}.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

  1. Break down the number −147-147 into its prime factors.
  2. Simplify the square root expression using the property of square roots.
  3. Simplify the square root of the negative number using the concept of imaginary numbers.
  4. Simplify the entire expression.

Common Mistakes

When simplifying the expression −147\sqrt{-147}, there are several common mistakes that students often make. These include:

  • Not breaking down the number under the square root sign into its prime factors.
  • Not using the property of square roots to separate the square root of a product into the product of the square roots.
  • Not simplifying the square root of the negative number using the concept of imaginary numbers.

Tips and Tricks

Here are some tips and tricks that can help you simplify the expression −147\sqrt{-147}:

  • Make sure to break down the number under the square root sign into its prime factors.
  • Use the property of square roots to separate the square root of a product into the product of the square roots.
  • Simplify the square root of the negative number using the concept of imaginary numbers.

Real-World Applications

The concept of simplifying the expression −147\sqrt{-147} has several real-world applications. For example, it can be used in the field of electrical engineering to analyze the behavior of electrical circuits. It can also be used in the field of physics to analyze the behavior of particles in a quantum system.

Conclusion

In conclusion, we have successfully simplified the expression −147\sqrt{-147} into its simplest radical form. We first broke down the number under the square root sign into its prime factors, then simplified the square root expression using the property of square roots. Finally, we simplified the square root of the negative number using the concept of imaginary numbers. The simplified expression is i3⋅7i\sqrt{3} \cdot 7.

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Introduction

In our previous article, we discussed how to simplify the expression −147\sqrt{-147} into its simplest radical form. We broke down the number under the square root sign into its prime factors, simplified the square root expression using the property of square roots, and finally simplified the square root of the negative number using the concept of imaginary numbers. In this article, we will answer some frequently asked questions about simplifying square roots of negative numbers.

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 finite decimal or fraction, such as 3 or 1/2. An imaginary number, on the other hand, is a number that can be expressed as the square root of a negative number, such as −1\sqrt{-1} or i3i\sqrt{3}.

Q: How do I simplify the square root of a negative number?

A: To simplify the square root of a negative number, you can use the property of square roots that allows you to separate the square root of a product into the product of the square roots. You can also use the concept of imaginary numbers to simplify the square root of a negative number.

Q: What is the imaginary unit ii?

A: The imaginary unit ii is a number that is defined as the square root of −1-1. It is used to extend the real number system to the complex number system.

Q: Can I simplify the square root of a negative number without using imaginary numbers?

A: No, you cannot simplify the square root of a negative number without using imaginary numbers. The concept of imaginary numbers is necessary to simplify the square root of a negative number.

Q: How do I know when to use the property of square roots to separate the square root of a product into the product of the square roots?

A: You should use the property of square roots to separate the square root of a product into the product of the square roots whenever you have a product under the square root sign.

Q: Can I simplify the square root of a negative number using a calculator?

A: Yes, you can simplify the square root of a negative number using a calculator. However, you should be aware that the calculator may not always give you the simplest radical form of the expression.

Q: What are some common mistakes to avoid when simplifying the square root of a negative number?

A: Some common mistakes to avoid when simplifying the square root of a negative number include:

  • Not breaking down the number under the square root sign into its prime factors.
  • Not using the property of square roots to separate the square root of a product into the product of the square roots.
  • Not simplifying the square root of the negative number using the concept of imaginary numbers.

Q: How do I check my work when simplifying the square root of a negative number?

A: To check your work when simplifying the square root of a negative number, you can plug the simplified expression back into the original expression and see if it is true. You can also use a calculator to check your work.

Q: Can I use the concept of imaginary numbers to simplify the square root of a negative number in a real-world application?

A: Yes, you can use the concept of imaginary numbers to simplify the square root of a negative number in a real-world application. For example, you can use the concept of imaginary numbers to analyze the behavior of electrical circuits or to analyze the behavior of particles in a quantum system.

Conclusion

In conclusion, we have answered some frequently asked questions about simplifying square roots of negative numbers. We have discussed the difference between real numbers and imaginary numbers, how to simplify the square root of a negative number, and how to check your work when simplifying the square root of a negative number. We have also discussed some common mistakes to avoid when simplifying the square root of a negative number and how to use the concept of imaginary numbers in a real-world application.

Final Answer

The final answer is i3⋅7\boxed{i\sqrt{3} \cdot 7}.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

  1. Break down the number under the square root sign into its prime factors.
  2. Simplify the square root expression using the property of square roots.
  3. Simplify the square root of the negative number using the concept of imaginary numbers.
  4. Check your work by plugging the simplified expression back into the original expression.

Common Mistakes

Here are some common mistakes to avoid when simplifying the square root of a negative number:

  • Not breaking down the number under the square root sign into its prime factors.
  • Not using the property of square roots to separate the square root of a product into the product of the square roots.
  • Not simplifying the square root of the negative number using the concept of imaginary numbers.

Tips and Tricks

Here are some tips and tricks that can help you simplify the square root of a negative number:

  • Make sure to break down the number under the square root sign into its prime factors.
  • Use the property of square roots to separate the square root of a product into the product of the square roots.
  • Simplify the square root of the negative number using the concept of imaginary numbers.

Real-World Applications

The concept of simplifying the square root of a negative number has several real-world applications. For example, it can be used in the field of electrical engineering to analyze the behavior of electrical circuits. It can also be used in the field of physics to analyze the behavior of particles in a quantum system.