Write The Radical As A Pure Imaginary Number.\[$\sqrt{-147} = \square\$\](Simplify Your Answer. Type An Exact Answer, Using Radicals As Needed. Express Complex Numbers In Terms Of \[$i\$\].)

Understanding Complex Numbers and Radicals

In mathematics, complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i^2 = -1. Radicals, on the other hand, are expressions that involve the square root of a number. When we encounter a negative number under a square root, we need to simplify it by expressing it as a pure imaginary number.

Expressing Negative Numbers as Pure Imaginary Numbers

To express a negative number as a pure imaginary number, we can use the fact that i^2 = -1. This means that we can rewrite a negative number as the square of an imaginary number. For example, -1 can be expressed as i^2.

Simplifying the Radical of -147

Now, let's simplify the radical of -147. We can start by expressing -147 as the square of an imaginary number. Since i^2 = -1, we can write:

$\sqrt{-147} = \sqrt{(-1)(147)}$

Using the Property of Radicals

We can use the property of radicals that states $\sqrt{ab} = \sqrt{a}\sqrt{b}$. Applying this property to the expression above, we get:

$\sqrt{-147} = \sqrt{(-1)(147)} = \sqrt{-1}\sqrt{147}$

Simplifying the Radical of 147

Now, let's simplify the radical of 147. We can start by finding the prime factorization of 147. The prime factorization of 147 is:

$147 = 3 \times 7^2$

Using the Property of Radicals

We can use the property of radicals that states $\sqrt{a^2} = a$. Applying this property to the expression above, we get:

$\sqrt{147} = \sqrt{3 \times 7^2} = 7\sqrt{3}$

Simplifying the Radical of -147

Now, let's simplify the radical of -147. We can substitute the expression for $\sqrt{147}$ that we found above:

$\sqrt{-147} = \sqrt{-1}\sqrt{147} = i(7\sqrt{3})$

Simplifying the Expression

We can simplify the expression by multiplying the terms:

$\sqrt{-147} = i(7\sqrt{3}) = 7i\sqrt{3}$

Conclusion

In conclusion, we have simplified the radical of -147 by expressing it as a pure imaginary number. We used the property of radicals and the fact that i^2 = -1 to simplify the expression. The final answer is:

$\boxed{7i\sqrt{3}}$

Final Answer

The final answer is $\boxed{7i\sqrt{3}}$.

Understanding Complex Numbers and Radicals

In mathematics, complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i^2 = -1. Radicals, on the other hand, are expressions that involve the square root of a number. When we encounter a negative number under a square root, we need to simplify it by expressing it as a pure imaginary number.

Expressing Negative Numbers as Pure Imaginary Numbers

To express a negative number as a pure imaginary number, we can use the fact that i^2 = -1. This means that we can rewrite a negative number as the square of an imaginary number. For example, -1 can be expressed as i^2.

Simplifying the Radical of -147

Now, let's simplify the radical of -147. We can start by expressing -147 as the square of an imaginary number. Since i^2 = -1, we can write:

$\sqrt{-147} = \sqrt{(-1)(147)}$

Using the Property of Radicals

We can use the property of radicals that states $\sqrt{ab} = \sqrt{a}\sqrt{b}$. Applying this property to the expression above, we get:

$\sqrt{-147} = \sqrt{(-1)(147)} = \sqrt{-1}\sqrt{147}$

Simplifying the Radical of 147

Now, let's simplify the radical of 147. We can start by finding the prime factorization of 147. The prime factorization of 147 is:

$147 = 3 \times 7^2$

Using the Property of Radicals

We can use the property of radicals that states $\sqrt{a^2} = a$. Applying this property to the expression above, we get:

$\sqrt{147} = \sqrt{3 \times 7^2} = 7\sqrt{3}$

Simplifying the Radical of -147

Now, let's simplify the radical of -147. We can substitute the expression for $\sqrt{147}$ that we found above:

$\sqrt{-147} = \sqrt{-1}\sqrt{147} = i(7\sqrt{3})$

Simplifying the Expression

We can simplify the expression by multiplying the terms:

$\sqrt{-147} = i(7\sqrt{3}) = 7i\sqrt{3}$

Conclusion

In conclusion, we have simplified the radical of -147 by expressing it as a pure imaginary number. We used the property of radicals and the fact that i^2 = -1 to simplify the expression. The final answer is:

$\boxed{7i\sqrt{3}}$

Q&A: Simplifying Radicals 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 without any imaginary unit, while an imaginary number is a number that can be expressed with the imaginary unit i.

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

A: To simplify the radical of a negative number, you can use the fact that i^2 = -1. This means that you can rewrite a negative number as the square of an imaginary number.

Q: What is the property of radicals that states $\sqrt{ab} = \sqrt{a}\sqrt{b}$?

A: This property states that the square root of a product of two numbers is equal to the product of the square roots of the two numbers.

Q: How do you find the prime factorization of a number?

A: To find the prime factorization of a number, you can start by dividing the number by the smallest prime number, which is 2. If the number is divisible by 2, you can continue dividing it by 2 until it is no longer divisible. Then, you can move on to the next prime number, which is 3, and repeat the process.

Q: What is the final answer to the problem of simplifying the radical of -147?

A: The final answer is $\boxed{7i\sqrt{3}}$.

Q: Can you provide more examples of simplifying radicals of negative numbers?

A: Yes, here are a few more examples:

• $\sqrt{-16} = \sqrt{(-1)(16)} = \sqrt{-1}\sqrt{16} = 4i$
• $\sqrt{-25} = \sqrt{(-1)(25)} = \sqrt{-1}\sqrt{25} = 5i$
• $\sqrt{-36} = \sqrt{(-1)(36)} = \sqrt{-1}\sqrt{36} = 6i$

I hope these examples help you understand how to simplify radicals of negative numbers.