Simplify: $\sqrt{-49}$. If The Result Is Not A Real Number, Enter $\varnothing$.Provide Your Answer Below:

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Introduction

When dealing with square roots, we are often asked to simplify expressions that involve both positive and negative numbers. In this case, we are tasked with simplifying the square root of -49. To approach this problem, we need to understand the properties of square roots and how they interact with negative 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, things get a bit more complicated.

The Nature of Negative Numbers

Negative numbers are the opposite of positive numbers. When we multiply two negative numbers together, we get a positive result. For example, -2 multiplied by -3 equals 6. However, when we take the square root of a negative number, we are essentially asking for a value that, when multiplied by itself, gives a negative result.

Simplifying the Square Root of -49

To simplify the square root of -49, we need to find two numbers whose product is -49. Since we are dealing with a negative number, we know that one of the numbers must be negative. Let's try to find two numbers whose product is -49.

Factoring -49

We can factor -49 as follows:

-49 = -1 × 49

Taking the Square Root

Now that we have factored -49, we can take the square root of each factor:

√(-1) × √(49)

Simplifying the Square Root of 49

The square root of 49 is 7, because 7 multiplied by 7 equals 49.

√(49) = 7

The Square Root of -1

The square root of -1 is a bit more complicated. In mathematics, we use the letter "i" to represent the square root of -1. This is because i multiplied by i equals -1.

√(-1) = i

Combining the Results

Now that we have simplified the square root of 49 and the square root of -1, we can combine the results:

√(-49) = √(-1) × √(49) = i × 7 = 7i

Conclusion

In conclusion, the square root of -49 is 7i. This is because the square root of -1 is i, and the square root of 49 is 7. When we multiply these two values together, we get 7i.

Final Answer

The final answer is: 7i\boxed{7i}

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Introduction

In our previous article, we explored the concept of simplifying the square root of a negative number. We learned that the square root of -49 is 7i, where i represents the square root of -1. In this article, we will answer some common questions related to simplifying the square root of a negative number.

Q&A

Q: What is the square root of -1?

A: The square root of -1 is represented by the letter "i". This is because i multiplied by i equals -1.

Q: Why do we use the letter "i" to represent the square root of -1?

A: We use the letter "i" to represent the square root of -1 because it is a convention in mathematics. This allows us to easily represent complex numbers and perform calculations involving square roots of negative numbers.

Q: Can the square root of a negative number be a real number?

A: No, the square root of a negative number cannot be a real number. This is because the product of two real numbers is always positive, and the square root of a negative number is essentially asking for a value that, when multiplied by itself, gives a negative result.

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

A: To simplify the square root of a negative number, you need to factor the number into two parts: one positive and one negative. Then, take the square root of each part and multiply them together.

Q: What is the difference between the square root of a negative number and a complex number?

A: The square root of a negative number is a complex number, but not all complex numbers are the square root of a negative number. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the square root of -1.

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 need to make sure that the calculator is set to display complex numbers. If the calculator is set to display only real numbers, it may not be able to simplify the square root of a negative number correctly.

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 factoring the number correctly
  • Not taking the square root of each part correctly
  • Not multiplying the square roots together correctly
  • Not using the correct notation for complex numbers

Conclusion

In conclusion, simplifying the square root of a negative number can be a bit tricky, but with the right techniques and notation, it can be done easily. By understanding the properties of square roots and complex numbers, you can simplify the square root of a negative number with confidence.

Final Tips

  • Make sure to factor the number correctly before taking the square root.
  • Use the correct notation for complex numbers.
  • Check your work carefully to avoid mistakes.
  • Practice simplifying the square root of negative numbers to become more comfortable with the concept.

Final Answer

The final answer is: 7i\boxed{7i}