Find The Product Of 4 ( 8 + 3 \sqrt{4}(\sqrt{8}+3 4 ​ ( 8 ​ + 3 ].A. 4 2 + 6 3 4 \sqrt{2}+6 \sqrt{3} 4 2 ​ + 6 3 ​ B. 2 3 + 6 2 \sqrt{3}+6 2 3 ​ + 6 C. 8 2 + 6 3 8 \sqrt{2}+6 \sqrt{3} 8 2 ​ + 6 3 ​ D. 4 2 + 6 4 \sqrt{2}+6 4 2 ​ + 6

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Understanding the Problem

The given problem involves finding the product of two square roots and a constant. To solve this, we need to apply the rules of multiplication and simplification of square roots. The expression $\sqrt{4}(\sqrt{8}+3)$ can be simplified by first evaluating the square roots and then multiplying the results.

Evaluating Square Roots

The square root of 4 is 2, as 2 multiplied by 2 equals 4. The square root of 8 can be simplified as $\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$.

Substituting Simplified Square Roots

Now that we have simplified the square roots, we can substitute these values back into the original expression. This gives us $2(2\sqrt{2}+3)$.

Multiplying the Expression

To find the product, we need to multiply the constant 2 by each term inside the parentheses. This results in $4\sqrt{2}+6$.

Comparing the Result with the Options

Now that we have found the product, we can compare it with the given options. The correct answer is $4\sqrt{2}+6$, which matches option D.

Conclusion

In this problem, we applied the rules of multiplication and simplification of square roots to find the product of $\sqrt{4}(\sqrt{8}+3)$. By simplifying the square roots and multiplying the expression, we arrived at the correct answer, which is $4\sqrt{2}+6$.

Final Answer

The final answer is option D, $4\sqrt{2}+6$.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Evaluate the square roots: $\sqrt{4} = 2$ and $\sqrt{8} = 2\sqrt{2}$.
2. Substitute the simplified square roots back into the original expression: $2(2\sqrt{2}+3)$.
3. Multiply the constant 2 by each term inside the parentheses: $4\sqrt{2}+6$.
4. Compare the result with the given options and select the correct answer.

Tips and Tricks

When working with square roots, it's essential to simplify them before multiplying or adding. This can help simplify the expression and make it easier to solve. Additionally, be careful when multiplying constants by terms inside parentheses, as this can affect the final result.

Common Mistakes

One common mistake when working with square roots is to forget to simplify them before multiplying or adding. This can lead to incorrect results and make it difficult to solve the problem. Another mistake is to multiply constants by terms inside parentheses incorrectly, which can also affect the final result.

Real-World Applications

The concept of simplifying square roots and multiplying expressions is essential in various real-world applications, such as physics, engineering, and computer science. Understanding how to simplify square roots and multiply expressions can help you solve complex problems and make informed decisions.

Practice Problems

Here are some practice problems to help you reinforce your understanding of simplifying square roots and multiplying expressions:

1. Simplify the expression $\sqrt{9}(\sqrt{16}+2)$.
2. Multiply the expression $3(\sqrt{25}+4)$.
3. Simplify the expression $\sqrt{36}(\sqrt{49}+3)$.

Conclusion

In conclusion, finding the product of $\sqrt{4}(\sqrt{8}+3)$ requires simplifying the square roots and multiplying the expression. By following the steps outlined in this solution, you can arrive at the correct answer, which is $4\sqrt{2}+6$. Remember to simplify square roots before multiplying or adding, and be careful when multiplying constants by terms inside parentheses. With practice and patience, you can become proficient in simplifying square roots and multiplying expressions.

Frequently Asked Questions

Q: What is the first step in simplifying the expression $\sqrt{4}(\sqrt{8}+3)$?

A: The first step is to evaluate the square roots. In this case, $\sqrt{4} = 2$ and $\sqrt{8} = 2\sqrt{2}$.

Q: How do I simplify the square root of 8?

A: The square root of 8 can be simplified as $\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$.

Q: What is the next step after simplifying the square roots?

A: After simplifying the square roots, we substitute the values back into the original expression. This gives us $2(2\sqrt{2}+3)$.

Q: How do I multiply the constant 2 by each term inside the parentheses?

A: To multiply the constant 2 by each term inside the parentheses, we multiply 2 by $2\sqrt{2}$ and 2 by 3. This results in $4\sqrt{2}+6$.

Q: What is the final answer to the expression $\sqrt{4}(\sqrt{8}+3)$?

A: The final answer is $4\sqrt{2}+6$.

Q: What are some common mistakes to avoid when simplifying square roots and multiplying expressions?

A: Some common mistakes to avoid include forgetting to simplify square roots before multiplying or adding, and multiplying constants by terms inside parentheses incorrectly.

Q: How do I apply the concept of simplifying square roots and multiplying expressions in real-world applications?

A: The concept of simplifying square roots and multiplying expressions is essential in various real-world applications, such as physics, engineering, and computer science. Understanding how to simplify square roots and multiply expressions can help you solve complex problems and make informed decisions.

Q: What are some practice problems to help me reinforce my understanding of simplifying square roots and multiplying expressions?

A: Here are some practice problems to help you reinforce your understanding:

1. Simplify the expression $\sqrt{9}(\sqrt{16}+2)$.
2. Multiply the expression $3(\sqrt{25}+4)$.
3. Simplify the expression $\sqrt{36}(\sqrt{49}+3)$.

Q: What are some tips and tricks for simplifying square roots and multiplying expressions?

A: Some tips and tricks include:

• Simplifying square roots before multiplying or adding
• Being careful when multiplying constants by terms inside parentheses
• Using the distributive property to multiply expressions
• Simplifying expressions by combining like terms

Q: How do I know if I have simplified the expression correctly?

A: To check if you have simplified the expression correctly, you can:

• Plug in the values into the original expression and simplify
• Use a calculator to check the result
• Compare your answer with the correct answer

Q: What are some common real-world applications of simplifying square roots and multiplying expressions?

A: Some common real-world applications include:

• Physics: Simplifying square roots and multiplying expressions is essential in solving problems related to motion, energy, and momentum.
• Engineering: Simplifying square roots and multiplying expressions is used in designing and building structures, such as bridges and buildings.
• Computer Science: Simplifying square roots and multiplying expressions is used in algorithms and data structures, such as sorting and searching.

Q: How can I practice simplifying square roots and multiplying expressions?

A: You can practice simplifying square roots and multiplying expressions by:

• Working on practice problems
• Using online resources and calculators
• Asking a teacher or tutor for help
• Joining a study group or online community

Q: What are some resources for learning more about simplifying square roots and multiplying expressions?

A: Some resources for learning more about simplifying square roots and multiplying expressions include:

• Online tutorials and videos
• Textbooks and workbooks
• Online communities and forums
• Teachers and tutors