Find The Sum: 4 3 + 11 12 4 \sqrt{3} + 11 \sqrt{12} 4 3 ​ + 11 12 ​ A. 15 15 15 \sqrt{15} 15 15 ​ B. 15 3 15 \sqrt{3} 15 3 ​ C. 26 3 26 \sqrt{3} 26 3 ​ D. 48 3 48 \sqrt{3} 48 3 ​

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Understanding the Problem

The problem requires finding the sum of two radical expressions: $4 \sqrt{3}$ and $11 \sqrt{12}$. To solve this, we need to first simplify the second radical expression by expressing it in terms of the first radical expression.

Simplifying the Second Radical Expression

We can simplify $\sqrt{12}$ by expressing it as a product of its prime factors. The prime factorization of 12 is $2^2 \times 3$. Therefore, we can rewrite $\sqrt{12}$ as $\sqrt{2^2 \times 3}$.

Applying the Property of Radicals

Using the property of radicals that $\sqrt{a^2} = a$, we can simplify $\sqrt{2^2 \times 3}$ as $2\sqrt{3}$.

Substituting the Simplified Expression

Now that we have simplified $\sqrt{12}$ as $2\sqrt{3}$, we can substitute this expression into the original problem: $4 \sqrt{3} + 11 \sqrt{12}$. This becomes $4 \sqrt{3} + 11 (2\sqrt{3})$.

Distributing the Coefficient

To simplify the expression further, we need to distribute the coefficient 11 to the radical expression $2\sqrt{3}$. This gives us $4 \sqrt{3} + 22\sqrt{3}$.

Combining Like Terms

Now that we have distributed the coefficient, we can combine the like terms in the expression. The two terms $4 \sqrt{3}$ and $22\sqrt{3}$ are like terms because they have the same radical expression $\sqrt{3}$. Therefore, we can combine them by adding their coefficients: $4 + 22 = 26$.

Simplifying the Expression

After combining the like terms, we are left with the expression $26\sqrt{3}$.

Conclusion

Therefore, the sum of $4 \sqrt{3}$ and $11 \sqrt{12}$ is $26\sqrt{3}$.

Final Answer

The final answer is $\boxed{26\sqrt{3}}$.

Comparison with the Options

Let's compare our final answer with the options provided:

• Option A: $15 \sqrt{15}$
• Option B: $15 \sqrt{3}$
• Option C: $26 \sqrt{3}$
• Option D: $48 \sqrt{3}$

Our final answer, $26\sqrt{3}$, matches option C.

Importance of Simplifying Radical Expressions

Simplifying radical expressions is an essential skill in mathematics, particularly in algebra and geometry. It allows us to rewrite complex expressions in a simpler form, making it easier to perform calculations and solve problems. In this case, simplifying the radical expression $\sqrt{12}$ as $2\sqrt{3}$ enabled us to combine like terms and arrive at the final answer.

Real-World Applications

Simplifying radical expressions has numerous real-world applications in fields such as engineering, physics, and computer science. For example, in engineering, simplifying radical expressions can help designers and architects create more efficient and cost-effective solutions. In physics, simplifying radical expressions can help scientists and researchers model complex phenomena and make predictions about the behavior of physical systems.

Tips for Simplifying Radical Expressions

Here are some tips for simplifying radical expressions:

• Express the radicand as a product of its prime factors: This can help you identify perfect squares and simplify the radical expression.
• Use the property of radicals that $\sqrt{a^2} = a$: This can help you simplify radical expressions by expressing them as a product of a perfect square and a radical expression.
• Distribute coefficients to the radical expression: This can help you simplify the expression by combining like terms.
• Combine like terms: This can help you simplify the expression by adding or subtracting the coefficients of like terms.

By following these tips, you can simplify radical expressions and arrive at the final answer with ease.

Conclusion

In conclusion, simplifying radical expressions is an essential skill in mathematics that has numerous real-world applications. By following the tips outlined in this article, you can simplify radical expressions and arrive at the final answer with ease. Remember to express the radicand as a product of its prime factors, use the property of radicals that $\sqrt{a^2} = a$, distribute coefficients to the radical expression, and combine like terms. With practice and patience, you can become proficient in simplifying radical expressions and tackle complex problems with confidence.

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Q&A: Simplifying Radical Expressions

In the previous article, we explored how to simplify the radical expression $\sqrt{12}$ as $2\sqrt{3}$ and arrived at the final answer of $26\sqrt{3}$. However, we received many questions from readers who were struggling to understand the concept of simplifying radical expressions. In this article, we will address some of the most frequently asked questions and provide additional tips and examples to help you master this essential skill.

Q: What is the difference between a radical expression and a simplified radical expression?

A: A radical expression is an expression that contains a radical symbol, such as $\sqrt{a}$ or $\sqrt{ab}$. A simplified radical expression is a radical expression that has been simplified by expressing the radicand as a product of its prime factors and using the property of radicals that $\sqrt{a^2} = a$.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, follow these steps:

1. Express the radicand as a product of its prime factors.
2. Identify perfect squares and simplify the radical expression using the property of radicals that $\sqrt{a^2} = a$.
3. Distribute coefficients to the radical expression.
4. Combine like terms.

Q: What is the property of radicals that $\sqrt{a^2} = a$?

A: The property of radicals that $\sqrt{a^2} = a$ states that the square root of a perfect square is equal to the number that is being squared. For example, $\sqrt{16} = 4$ because $4^2 = 16$.

Q: How do I distribute coefficients to the radical expression?

A: To distribute coefficients to the radical expression, multiply the coefficient by the radical expression. For example, if we have the expression $4\sqrt{3}$, we can distribute the coefficient 4 to the radical expression $\sqrt{3}$ by multiplying 4 by $\sqrt{3}$, which gives us $4\sqrt{3}$.

Q: What is the difference between combining like terms and simplifying a radical expression?

A: Combining like terms involves adding or subtracting the coefficients of like terms, whereas simplifying a radical expression involves expressing the radicand as a product of its prime factors and using the property of radicals that $\sqrt{a^2} = a$.

Q: Can you provide more examples of simplifying radical expressions?

A: Here are a few more examples:

• $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$
• $\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$
• $\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$

Q: How do I know when to simplify a radical expression?

A: You should simplify a radical expression whenever possible. Simplifying radical expressions can help you:

• Make calculations easier
• Solve problems more efficiently
• Understand complex concepts better

Q: Can you provide additional tips for simplifying radical expressions?

A: Here are a few more tips:

• Practice, practice, practice! The more you practice simplifying radical expressions, the more comfortable you will become with the process.
• Use a calculator to check your work. This can help you identify any mistakes and ensure that your answer is correct.
• Break down complex problems into smaller, more manageable parts. This can help you simplify radical expressions and arrive at the final answer with ease.

Conclusion

In conclusion, simplifying radical expressions is an essential skill in mathematics that has numerous real-world applications. By following the tips outlined in this article, you can simplify radical expressions and arrive at the final answer with ease. Remember to express the radicand as a product of its prime factors, use the property of radicals that $\sqrt{a^2} = a$, distribute coefficients to the radical expression, and combine like terms. With practice and patience, you can become proficient in simplifying radical expressions and tackle complex problems with confidence.