Simplify: $\sqrt{-49}$

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Introduction

Understanding Square Roots Square roots are a fundamental concept in mathematics, used to find the number that, when multiplied by itself, gives a specified value. In this case, we are tasked with simplifying the expression $\sqrt{-49}$. To do this, we need to understand the properties of square roots and how to handle negative numbers.

Properties of Square Roots

• Definition: The square root of a number $a$ is a value $x$ such that $x^2 = a$.
• Positive and Negative Square Roots: Every positive number has two square roots, one positive and one negative. For example, the square roots of 16 are 4 and -4.
• Square Root of Negative Numbers: The square root of a negative number is an imaginary number, which is a complex number that cannot be represented on the real number line.

Simplifying $\sqrt{-49}$

To simplify $\sqrt{-49}$, we need to find the number that, when multiplied by itself, gives -49. Since -49 is a negative number, we know that its square root will be an imaginary number.

Step 1: Factorize -49

We can start by factorizing -49 into its prime factors. We know that -49 can be written as $-1 \times 49$. Since 49 is a perfect square, we can further simplify it as $7^2$.

Step 2: Simplify the Square Root

Now that we have factorized -49, we can simplify the square root. We know that the square root of a product is equal to the product of the square roots. Therefore, we can write:

$$\sqrt{-49} = \sqrt{-1 \times 49} = \sqrt{-1} \times \sqrt{49}$$

Step 3: Simplify the Imaginary Unit

The square root of -1 is an imaginary unit, denoted by $i$. Therefore, we can simplify the expression as:

$$\sqrt{-49} = i \times \sqrt{49}$$

Step 4: Simplify the Square Root of 49

We know that the square root of 49 is 7. Therefore, we can simplify the expression as:

$$\sqrt{-49} = i \times 7$$

Step 5: Simplify the Expression

Finally, we can simplify the expression by multiplying the imaginary unit $i$ by 7. This gives us:

$$\sqrt{-49} = 7i$$

Conclusion

In this article, we simplified the expression $\sqrt{-49}$ by using the properties of square roots and the definition of imaginary numbers. We factorized -49 into its prime factors, simplified the square root, and finally obtained the simplified expression $7i$.

Frequently Asked Questions

• What is the square root of -49? The square root of -49 is $7i$.
• Is the square root of -49 a real number? No, the square root of -49 is an imaginary number.
• Can the square root of -49 be simplified further? No, the square root of -49 cannot be simplified further.

Final Thoughts

Simplifying the expression $\sqrt{-49}$ requires a good understanding of the properties of square roots and imaginary numbers. By following the steps outlined in this article, we can simplify the expression and obtain the final result $7i$. This is an important concept in mathematics, and it has many applications in fields such as physics and engineering.

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Introduction

In our previous article, we simplified the expression $\sqrt{-49}$ by using the properties of square roots and the definition of imaginary numbers. In this article, we will answer some frequently asked questions related to the simplification of $\sqrt{-49}$.

Q&A

Q: What is the square root of -49?

A: The square root of -49 is $7i$.

Q: Is the square root of -49 a real number?

A: No, the square root of -49 is an imaginary number.

Q: Can the square root of -49 be simplified further?

A: No, the square root of -49 cannot be simplified further.

Q: Why is the square root of -49 an imaginary number?

A: The square root of -49 is an imaginary number because -49 is a negative number. The square root of a negative number is always an imaginary number.

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

A: A real number is a number that can be represented on the real number line, whereas an imaginary number is a complex number that cannot be represented on the real number line.

Q: Can you give an example of a real number and an imaginary number?

A: Yes, 4 is a real number, whereas $4i$ is an imaginary number.

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

A: To simplify the square root of a negative number, you need to factorize the number into its prime factors and then simplify the square root.

Q: Can you give an example of simplifying the square root of a negative number?

A: Yes, the square root of -49 can be simplified as follows:

$$\sqrt{-49} = \sqrt{-1 \times 49} = \sqrt{-1} \times \sqrt{49} = i \times 7 = 7i$$

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

A: The imaginary unit $i$ is a complex number that satisfies the equation $i^2 = -1$. It is used to represent imaginary numbers.

Q: Can you give an example of using the imaginary unit $i$?

A: Yes, the square root of -49 can be simplified using the imaginary unit $i$ as follows:

$$\sqrt{-49} = i \times \sqrt{49} = i \times 7 = 7i$$

Conclusion

In this article, we answered some frequently asked questions related to the simplification of $\sqrt{-49}$. We explained the properties of square roots and imaginary numbers, and provided examples of simplifying the square root of a negative number.

Final Thoughts

Simplifying the expression $\sqrt{-49}$ requires a good understanding of the properties of square roots and imaginary numbers. By following the steps outlined in this article, we can simplify the expression and obtain the final result $7i$. This is an important concept in mathematics, and it has many applications in fields such as physics and engineering.

Additional Resources

• Introduction to Square Roots
• Introduction to Imaginary Numbers
• Simplifying Square Roots

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