Combining Like Terms with Radicals

Oct 13, 2025 by miodeya.com 95 views

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Introduction

When dealing with mathematical expressions that involve radicals, it's essential to understand how to simplify them. In this article, we will focus on simplifying an expression that contains radicals, specifically square roots. We will explore the concept of combining like terms with radicals and provide a step-by-step guide on how to simplify the given expression.

Understanding Radicals

Radicals are mathematical expressions that involve the square root of a number. They are denoted by the symbol √ and are used to represent the positive square root of a number. For example, √4 = 2, since 2 multiplied by 2 equals 4. Radicals can also be used to represent negative square roots, which are denoted by the symbol -√.

Combining Like Terms with Radicals

When simplifying an expression that contains radicals, it's essential to combine like terms. Like terms are terms that have the same variable raised to the same power. In the case of radicals, like terms are terms that have the same radical expression. For example, 2√5 and 3√5 are like terms because they both have the radical expression √5.

Simplifying the Expression

The given expression is:

${ -\sqrt{5} - 2\sqrt{5} - \sqrt{2} }$

To simplify this expression, we need to combine the like terms. The like terms in this expression are the terms that have the radical expression √5. We can combine these terms by adding their coefficients.

Step 1: Identify the Like Terms

The like terms in the given expression are:

-√5
-2√5

These terms have the same radical expression, which is √5.

Step 2: Combine the Like Terms

To combine the like terms, we need to add their coefficients. The coefficient of -√5 is -1, and the coefficient of -2√5 is -2. We can add these coefficients by adding their absolute values and keeping the same sign.

Step 3: Simplify the Expression

Using the steps above, we can simplify the expression as follows:

${ -\sqrt{5} - 2\sqrt{5} - \sqrt{2} }$

= -3√5 - √2

Conclusion

In this article, we simplified an expression that contained radicals by combining like terms. We identified the like terms, combined their coefficients, and simplified the expression. By following these steps, we can simplify any expression that contains radicals.

Frequently Asked Questions

Q: What are like terms in the context of radicals?

A: Like terms in the context of radicals are terms that have the same radical expression.

Q: How do I combine like terms with radicals?

A: To combine like terms with radicals, you need to add their coefficients.

Q: What is the simplified form of the given expression?

A: The simplified form of the given expression is -3√5 - √2.

Final Thoughts

Simplifying expressions that contain radicals is an essential skill in mathematics. By understanding how to combine like terms with radicals, we can simplify complex expressions and solve mathematical problems more efficiently. In this article, we provided a step-by-step guide on how to simplify the given expression and answered frequently asked questions related to the topic.

Additional Resources

For more information on simplifying expressions that contain radicals, we recommend the following resources:

Khan Academy: Simplifying Radical Expressions
Mathway: Simplifying Radical Expressions
Wolfram Alpha: Simplifying Radical Expressions

By following these resources and practicing the steps outlined in this article, you can become proficient in simplifying expressions that contain radicals.

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Introduction

In our previous article, we discussed how to simplify expressions that contain radicals by combining like terms. In this article, we will answer some frequently asked questions related to simplifying expressions with radicals.

Q&A

Q: What are like terms in the context of radicals?

A: Like terms in the context of radicals are terms that have the same radical expression. For example, 2√5 and 3√5 are like terms because they both have the radical expression √5.

Q: How do I combine like terms with radicals?

A: To combine like terms with radicals, you need to add their coefficients. For example, to combine 2√5 and 3√5, you would add their coefficients: 2 + 3 = 5. The resulting term would be 5√5.

Q: What is the simplified form of the expression -\sqrt{5} - 2\sqrt{5} - \sqrt{2}?

A: The simplified form of the expression -\sqrt{5} - 2\sqrt{5} - \sqrt{2} is -3\sqrt{5} - \sqrt{2}.

Q: Can I simplify an expression that contains a negative radical?

A: Yes, you can simplify an expression that contains a negative radical. To simplify a negative radical, you need to multiply the radical by -1. For example, to simplify -\sqrt5}, you would multiply the radical by -1 -\sqrt{5 = -1 \times \sqrt{5}.

Q: How do I simplify an expression that contains multiple radicals?

A: To simplify an expression that contains multiple radicals, you need to combine like terms and then simplify the resulting expression. For example, to simplify the expression \sqrt5} + 2\sqrt{5} + \sqrt{2}, you would first combine the like terms \sqrt{5 + 2\sqrt5} = 3\sqrt{5}. Then, you would simplify the resulting expression 3\sqrt{5 + \sqrt{2}.

Q: Can I simplify an expression that contains a rational exponent?

A: Yes, you can simplify an expression that contains a rational exponent. To simplify a rational exponent, you need to rewrite the exponent as a radical. For example, to simplify the expression x^1/2}, you would rewrite the exponent as a radical x^{1/2 = \sqrt{x}.

Q: How do I simplify an expression that contains a radical and a rational number?

A: To simplify an expression that contains a radical and a rational number, you need to multiply the radical by the rational number. For example, to simplify the expression 2\sqrt5}, you would multiply the radical by 2 2\sqrt{5 = 2 \times \sqrt{5}.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions with radicals. We discussed how to combine like terms, simplify negative radicals, and simplify expressions that contain multiple radicals, rational exponents, and rational numbers.

Final Thoughts

Simplifying expressions that contain radicals is an essential skill in mathematics. By understanding how to combine like terms, simplify negative radicals, and simplify expressions that contain multiple radicals, rational exponents, and rational numbers, you can simplify complex expressions and solve mathematical problems more efficiently.

Additional Resources

For more information on simplifying expressions that contain radicals, we recommend the following resources:

Khan Academy: Simplifying Radical Expressions
Mathway: Simplifying Radical Expressions
Wolfram Alpha: Simplifying Radical Expressions

By following these resources and practicing the steps outlined in this article, you can become proficient in simplifying expressions that contain radicals.