Perform The Indicated Operations And Simplify.1. Add: { (-2+\sqrt -144})+(20-\sqrt{-49})$}$ Sum = { \square$}$2. Subtract { (-2+\sqrt{-144 )-(20-\sqrt{-49})$}$ Difference = { \square$}$

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, focusing on the given problem of adding and subtracting expressions containing square roots. We will break down the steps involved in simplifying these expressions and provide a clear understanding of the concepts.

Understanding Square Roots

Before we dive into the problem, let's briefly review the concept of 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. We denote the square root of a number using the symbol √.

Simplifying Radical Expressions

Now, let's move on to the problem at hand. We are given two expressions to simplify:

  1. Add: {(-2+\sqrt{-144})+(20-\sqrt{-49})$}$
  2. Subtract: {(-2+\sqrt{-144})-(20-\sqrt{-49})$}$

To simplify these expressions, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expressions inside the parentheses.
  2. Simplify the square roots.
  3. Combine like terms.

Step 1: Evaluate the Expressions Inside the Parentheses

Let's start by evaluating the expressions inside the parentheses:

{(-2+\sqrt{-144})+(20-\sqrt{-49})$}$

We can simplify the expressions inside the parentheses by evaluating the square roots:

{(-2+\sqrt{-144}) = -2 + \sqrt{(-1)(144)} = -2 + \sqrt{-1}\sqrt{144} = -2 + 12i$}$

{(20-\sqrt{-49}) = 20 - \sqrt{(-1)(49)} = 20 - \sqrt{-1}\sqrt{49} = 20 - 7i$}$

Now, we can substitute these simplified expressions back into the original equation:

{(-2+\sqrt{-144})+(20-\sqrt{-49}) = (-2 + 12i) + (20 - 7i)$}$

Step 2: Simplify the Square Roots

Next, we need to simplify the square roots. We can do this by factoring the radicand (the number inside the square root):

{\sqrt{-144} = \sqrt{(-1)(144)} = \sqrt{-1}\sqrt{144} = 12i$}$

{\sqrt{-49} = \sqrt{(-1)(49)} = \sqrt{-1}\sqrt{49} = 7i$}$

Now, we can substitute these simplified square roots back into the original equation:

{(-2+\sqrt{-144})+(20-\sqrt{-49}) = (-2 + 12i) + (20 - 7i)$}$

Step 3: Combine Like Terms

Finally, we can combine like terms:

{(-2 + 12i) + (20 - 7i) = -2 + 20 + 12i - 7i = 18 + 5i$}$

Therefore, the sum of the two expressions is:

Sum = 18 + 5i

Step 4: Subtract the Expressions

Now, let's move on to the second expression:

{(-2+\sqrt{-144})-(20-\sqrt{-49})$}$

We can simplify this expression by following the same steps as before:

{(-2+\sqrt{-144}) = -2 + \sqrt{(-1)(144)} = -2 + \sqrt{-1}\sqrt{144} = -2 + 12i$}$

{(20-\sqrt{-49}) = 20 - \sqrt{(-1)(49)} = 20 - \sqrt{-1}\sqrt{49} = 20 - 7i$}$

Now, we can substitute these simplified expressions back into the original equation:

{(-2+\sqrt{-144})-(20-\sqrt{-49}) = (-2 + 12i) - (20 - 7i)$}$

Step 5: Simplify the Square Roots

Next, we need to simplify the square roots. We can do this by factoring the radicand (the number inside the square root):

{\sqrt{-144} = \sqrt{(-1)(144)} = \sqrt{-1}\sqrt{144} = 12i$}$

{\sqrt{-49} = \sqrt{(-1)(49)} = \sqrt{-1}\sqrt{49} = 7i$}$

Now, we can substitute these simplified square roots back into the original equation:

{(-2+\sqrt{-144})-(20-\sqrt{-49}) = (-2 + 12i) - (20 - 7i)$}$

Step 6: Combine Like Terms

Finally, we can combine like terms:

{(-2 + 12i) - (20 - 7i) = -2 - 20 + 12i + 7i = -22 + 19i$}$

Therefore, the difference of the two expressions is:

Difference = -22 + 19i

Conclusion

In this article, we have explored the process of simplifying radical expressions, focusing on the given problem of adding and subtracting expressions containing square roots. We have broken down the steps involved in simplifying these expressions and provided a clear understanding of the concepts. By following the order of operations (PEMDAS) and simplifying the square roots, we have arrived at the final answers for the sum and difference of the two expressions.

Key Takeaways

  • Simplifying radical expressions involves following the order of operations (PEMDAS).
  • We need to evaluate the expressions inside the parentheses, simplify the square roots, and combine like terms.
  • The sum of the two expressions is 18 + 5i, and the difference of the two expressions is -22 + 19i.

Final Thoughts

Introduction

In our previous article, we explored the process of simplifying radical expressions, focusing on the given problem of adding and subtracting expressions containing square roots. We have broken down the steps involved in simplifying these expressions and provided a clear understanding of the concepts. In this article, we will answer some frequently asked questions related to simplifying radical expressions.

Q&A

Q: What is the difference between a radical and a square root?

A: A radical is a mathematical expression that involves a square root, while a square root is a specific type of radical that represents the number that, when multiplied by itself, gives the original number.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to follow the order of operations (PEMDAS):

  1. Evaluate the expressions inside the parentheses.
  2. Simplify the square roots.
  3. Combine like terms.

Q: What is the rule for simplifying square roots?

A: The rule for simplifying square roots is to factor the radicand (the number inside the square root) and simplify the expression.

Q: How do I simplify a radical expression with a negative number inside the square root?

A: To simplify a radical expression with a negative number inside the square root, you need to follow the same steps as before:

  1. Factor the radicand.
  2. Simplify the expression.
  3. Combine like terms.

Q: Can I simplify a radical expression with a variable inside the square root?

A: Yes, you can simplify a radical expression with a variable inside the square root. However, you need to follow the same steps as before:

  1. Factor the radicand.
  2. Simplify the expression.
  3. Combine like terms.

Q: How do I add and subtract radical expressions?

A: To add and subtract radical expressions, you need to follow the same steps as before:

  1. Simplify the expressions inside the parentheses.
  2. Combine like terms.

Q: Can I simplify a radical expression with a complex number inside the square root?

A: Yes, you can simplify a radical expression with a complex number inside the square root. However, you need to follow the same steps as before:

  1. Factor the radicand.
  2. Simplify the expression.
  3. Combine like terms.

Q: How do I simplify a radical expression with a rational number inside the square root?

A: To simplify a radical expression with a rational number inside the square root, you need to follow the same steps as before:

  1. Factor the radicand.
  2. Simplify the expression.
  3. Combine like terms.

Q: Can I simplify a radical expression with a negative rational number inside the square root?

A: Yes, you can simplify a radical expression with a negative rational number inside the square root. However, you need to follow the same steps as before:

  1. Factor the radicand.
  2. Simplify the expression.
  3. Combine like terms.

Conclusion

In this article, we have answered some frequently asked questions related to simplifying radical expressions. We have provided a clear understanding of the concepts involved and have outlined the steps to simplify radical expressions. By following the order of operations (PEMDAS) and simplifying the square roots, you can simplify radical expressions with ease.

Key Takeaways

  • Simplifying radical expressions involves following the order of operations (PEMDAS).
  • We need to evaluate the expressions inside the parentheses, simplify the square roots, and combine like terms.
  • We can simplify radical expressions with negative numbers, variables, complex numbers, rational numbers, and negative rational numbers.

Final Thoughts

Simplifying radical expressions is an essential skill in mathematics, and it requires a clear understanding of the concepts involved. By following the steps outlined in this article, you can simplify radical expressions with ease and arrive at the final answers. Remember to always follow the order of operations (PEMDAS) and simplify the square roots to ensure accurate results.