Simplify:${ 2 \sqrt{27} - \sqrt{3} }$A. { 5 \sqrt{3} $}$ B. { 17 \sqrt{3} $}$ C. { \sqrt{24} $}$ D. { 2 \sqrt{24} $}$

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Understanding the Problem

The given expression involves the simplification of a square root expression. We are required to simplify the expression $2 \sqrt{27} - \sqrt{3}$ and choose the correct answer from the options provided.

Breaking Down the Expression

To simplify the expression, we need to break it down into its prime factors. The expression $2 \sqrt{27}$ can be simplified as follows:

$$2 \sqrt{27} = 2 \sqrt{9 \times 3} = 2 \times 3 \times \sqrt{3} = 6 \sqrt{3}$$

Simplifying the Expression

Now that we have simplified the first part of the expression, we can substitute it back into the original expression:

$$2 \sqrt{27} - \sqrt{3} = 6 \sqrt{3} - \sqrt{3}$$

Combining Like Terms

We can combine the like terms in the expression:

$$6 \sqrt{3} - \sqrt{3} = 5 \sqrt{3}$$

Conclusion

Therefore, the simplified expression is $5 \sqrt{3}$.

Choosing the Correct Answer

Based on our simplification, we can see that the correct answer is option A: $5 \sqrt{3}$.

Why the Other Options are Incorrect

Let's analyze why the other options are incorrect:

• Option B: $17 \sqrt{3}$ is incorrect because our simplification resulted in $5 \sqrt{3}$, not $17 \sqrt{3}$.
• Option C: $\sqrt{24}$ is incorrect because our simplification resulted in $5 \sqrt{3}$, not $\sqrt{24}$.
• Option D: $2 \sqrt{24}$ is incorrect because our simplification resulted in $5 \sqrt{3}$, not $2 \sqrt{24}$.

Final Answer

The final answer is option A: $5 \sqrt{3}$.

Additional Tips and Tricks

• When simplifying square root expressions, it's essential to break them down into their prime factors.
• Combining like terms can help simplify the expression further.
• Make sure to check your work and verify that your answer is correct.

Common Mistakes to Avoid

• Failing to break down the expression into its prime factors.
• Not combining like terms.
• Not verifying the answer.

Real-World Applications

• Simplifying square root expressions is essential in various mathematical and scientific applications, such as physics, engineering, and computer science.
• Understanding how to simplify square root expressions can help you solve complex problems and make informed decisions.

Conclusion

In conclusion, simplifying the expression $2 \sqrt{27} - \sqrt{3}$ requires breaking it down into its prime factors, combining like terms, and verifying the answer. The correct answer is option A: $5 \sqrt{3}$.

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Frequently Asked Questions

Q: What is the prime factorization of 27?

A: The prime factorization of 27 is $3^3$.

Q: How do I simplify the expression $2 \sqrt{27}$?

A: To simplify the expression $2 \sqrt{27}$, you need to break it down into its prime factors. The prime factorization of 27 is $3^3$, so you can rewrite the expression as $2 \sqrt{3^3}$. Then, you can simplify it further by taking the square root of the prime factors: $2 \times 3 \times \sqrt{3} = 6 \sqrt{3}$.

Q: How do I combine like terms in the expression $6 \sqrt{3} - \sqrt{3}$?

A: To combine like terms in the expression $6 \sqrt{3} - \sqrt{3}$, you need to identify the like terms, which are the terms with the same variable and exponent. In this case, the like terms are the terms with the variable $\sqrt{3}$. You can combine these terms by subtracting the coefficients: $6 \sqrt{3} - \sqrt{3} = 5 \sqrt{3}$.

Q: Why is the expression $2 \sqrt{24}$ not a correct answer?

A: The expression $2 \sqrt{24}$ is not a correct answer because it does not match the simplified expression $5 \sqrt{3}$. The expression $2 \sqrt{24}$ can be simplified further by breaking down the prime factors of 24, which is $2^3 \times 3$. Then, you can rewrite the expression as $2 \times 2 \times \sqrt{2 \times 3} = 4 \sqrt{6}$. This is not equal to $5 \sqrt{3}$.

Q: What are some common mistakes to avoid when simplifying square root expressions?

A: Some common mistakes to avoid when simplifying square root expressions include:

• Failing to break down the expression into its prime factors.
• Not combining like terms.
• Not verifying the answer.

Q: How do I verify the answer?

A: To verify the answer, you need to check that the simplified expression matches one of the options. In this case, the simplified expression is $5 \sqrt{3}$, which matches option A.

Q: What are some real-world applications of simplifying square root expressions?

A: Simplifying square root expressions is essential in various mathematical and scientific applications, such as physics, engineering, and computer science. Understanding how to simplify square root expressions can help you solve complex problems and make informed decisions.

Q: Can I use a calculator to simplify square root expressions?

A: While a calculator can be useful for simplifying square root expressions, it's not always the best approach. A calculator may not always provide the simplest form of the expression, and it may not be able to handle complex expressions. It's often better to simplify the expression by hand using the rules of algebra.

Q: How do I know when to use the square root symbol?

A: The square root symbol is used to indicate the positive square root of a number. If you're unsure whether to use the square root symbol or not, try to simplify the expression by hand using the rules of algebra. If the expression can be simplified further, then you may need to use the square root symbol.

Q: Can I simplify square root expressions with negative numbers?

A: Yes, you can simplify square root expressions with negative numbers. However, you need to be careful when working with negative numbers, as they can affect the sign of the expression. In general, the square root of a negative number is an imaginary number, which is denoted by the letter $i$.