Simplify The Expression:${ 15 \sqrt{3} + 10 \sqrt{7} + 5 \sqrt{3} + 20 \sqrt{7} = }$

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Introduction

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When dealing with algebraic expressions that involve radicals, it's essential to understand how to simplify them by combining like terms. In this discussion, we will focus on simplifying an expression that contains radicals, specifically square roots of numbers. We will use the given expression as an example and demonstrate the step-by-step process of simplifying it.

The Given Expression

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The given expression is:

${ 15 \sqrt{3} + 10 \sqrt{7} + 5 \sqrt{3} + 20 \sqrt{7} }$

Understanding Like Terms

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Like terms are terms that have the same variable raised to the same power. In the context of radicals, like terms are terms that have the same radical expression. In this case, we have two sets of like terms: $15 \sqrt{3}$ and $5 \sqrt{3}$, and $10 \sqrt{7}$ and $20 \sqrt{7}$.

Combining Like Terms

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To simplify the expression, we need to combine the like terms. We can do this by adding or subtracting the coefficients of the like terms. In this case, we will add the coefficients of the like terms.

Combining the Like Terms with $\sqrt{3}$

The like terms with $\sqrt{3}$ are $15 \sqrt{3}$ and $5 \sqrt{3}$. We can combine these terms by adding their coefficients:

${ 15 \sqrt{3} + 5 \sqrt{3} = (15 + 5) \sqrt{3} = 20 \sqrt{3} }$

Combining the Like Terms with $\sqrt{7}$

The like terms with $\sqrt{7}$ are $10 \sqrt{7}$ and $20 \sqrt{7}$. We can combine these terms by adding their coefficients:

${ 10 \sqrt{7} + 20 \sqrt{7} = (10 + 20) \sqrt{7} = 30 \sqrt{7} }$

Simplifying the Expression

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Now that we have combined the like terms, we can simplify the expression by adding the simplified terms:

${ 20 \sqrt{3} + 30 \sqrt{7} }$

This is the simplified expression.

Conclusion

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In this discussion, we simplified an expression that contained radicals by combining like terms. We identified the like terms, combined them by adding their coefficients, and simplified the expression. This process is essential in algebra and is used to simplify complex expressions.

Example Use Case

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Simplifying expressions with radicals is a common task in algebra and is used in various applications, such as:

• Calculating distances and lengths in geometry
• Solving equations and inequalities in algebra
• Working with trigonometry and calculus

By understanding how to simplify expressions with radicals, you can solve problems more efficiently and effectively.

Tips and Tricks

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When simplifying expressions with radicals, remember to:

• Identify the like terms
• Combine the like terms by adding their coefficients
• Simplify the expression by adding the simplified terms

By following these steps, you can simplify expressions with radicals and solve problems more efficiently.

Common Mistakes

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When simplifying expressions with radicals, common mistakes include:

• Failing to identify the like terms
• Combining the like terms incorrectly
• Simplifying the expression incorrectly

To avoid these mistakes, make sure to carefully identify the like terms, combine them correctly, and simplify the expression accurately.

Final Thoughts

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Simplifying expressions with radicals is an essential skill in algebra and is used in various applications. By understanding how to combine like terms and simplify expressions, you can solve problems more efficiently and effectively. Remember to identify the like terms, combine them correctly, and simplify the expression accurately to avoid common mistakes.

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Introduction

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In our previous discussion, we simplified an expression that contained radicals by combining like terms. In this Q&A article, we will address some common questions and provide additional examples to help you better understand how to simplify expressions with radicals.

Q&A

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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, $15 \sqrt{3}$ and $5 \sqrt{3}$ are like terms because they both have the radical expression $\sqrt{3}$.

Q: How do I identify like terms in an expression?

A: To identify like terms in an expression, look for terms that have the same radical expression. You can also use the distributive property to rewrite the expression and make it easier to identify the like terms.

Q: Can I combine like terms with different coefficients?

A: Yes, you can combine like terms with different coefficients. To do this, add the coefficients of the like terms. For example, $15 \sqrt{3} + 5 \sqrt{3} = (15 + 5) \sqrt{3} = 20 \sqrt{3}$.

Q: What if I have like terms with different variables?

A: If you have like terms with different variables, you cannot combine them. For example, $15 \sqrt{3}$ and $10 \sqrt{7}$ are not like terms because they have different variables.

Q: Can I simplify an expression with radicals by combining like terms and then simplifying the resulting expression?

A: Yes, you can simplify an expression with radicals by combining like terms and then simplifying the resulting expression. This is a common technique used in algebra to simplify complex expressions.

Example 1: Simplifying an Expression with Radicals

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Simplify the expression:

${ 12 \sqrt{2} + 8 \sqrt{2} + 15 \sqrt{5} + 20 \sqrt{5} }$

Step 1: Identify the like terms

The like terms in this expression are $12 \sqrt{2}$ and $8 \sqrt{2}$, and $15 \sqrt{5}$ and $20 \sqrt{5}$.

Step 2: Combine the like terms

Combine the like terms by adding their coefficients:

${ 12 \sqrt{2} + 8 \sqrt{2} = (12 + 8) \sqrt{2} = 20 \sqrt{2} }$

${ 15 \sqrt{5} + 20 \sqrt{5} = (15 + 20) \sqrt{5} = 35 \sqrt{5} }$

Step 3: Simplify the resulting expression

The simplified expression is:

${ 20 \sqrt{2} + 35 \sqrt{5} }$

Example 2: Simplifying an Expression with Radicals

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Simplify the expression:

${ 25 \sqrt{3} + 10 \sqrt{3} + 15 \sqrt{6} + 20 \sqrt{6} }$

Step 1: Identify the like terms

The like terms in this expression are $25 \sqrt{3}$ and $10 \sqrt{3}$, and $15 \sqrt{6}$ and $20 \sqrt{6}$.

Step 2: Combine the like terms

Combine the like terms by adding their coefficients:

${ 25 \sqrt{3} + 10 \sqrt{3} = (25 + 10) \sqrt{3} = 35 \sqrt{3} }$

${ 15 \sqrt{6} + 20 \sqrt{6} = (15 + 20) \sqrt{6} = 35 \sqrt{6} }$

Step 3: Simplify the resulting expression

The simplified expression is:

${ 35 \sqrt{3} + 35 \sqrt{6} }$

Conclusion

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In this Q&A article, we addressed some common questions and provided additional examples to help you better understand how to simplify expressions with radicals. Remember to identify the like terms, combine them correctly, and simplify the resulting expression to avoid common mistakes.

Tips and Tricks

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When simplifying expressions with radicals, remember to:

• Identify the like terms
• Combine the like terms by adding their coefficients
• Simplify the resulting expression

By following these steps, you can simplify expressions with radicals and solve problems more efficiently.

Common Mistakes

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When simplifying expressions with radicals, common mistakes include:

• Failing to identify the like terms
• Combining the like terms incorrectly
• Simplifying the resulting expression incorrectly

To avoid these mistakes, make sure to carefully identify the like terms, combine them correctly, and simplify the resulting expression accurately.

Final Thoughts

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Simplifying expressions with radicals is an essential skill in algebra and is used in various applications. By understanding how to combine like terms and simplify expressions, you can solve problems more efficiently and effectively. Remember to identify the like terms, combine them correctly, and simplify the resulting expression accurately to avoid common mistakes.