Select The Correct Answer.What Is 343 \sqrt{343} 343 ​ In Simplest Form?A. 7 49 7 \sqrt{49} 7 49 ​ B. 7 7 7 \sqrt{7} 7 7 ​ C. 7 D. 49 7 49 \sqrt{7} 49 7 ​

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Understanding Square Roots

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A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. In mathematical notation, this is represented as $\sqrt{16} = 4$.

Simplifying Square Roots: The Basics

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To simplify a square root, we need to find the largest perfect square that divides the number inside the square root. A perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 is a perfect square because it can be expressed as $4 \times 4$, while 17 is not a perfect square because it cannot be expressed as the product of an integer with itself.

Simplifying $\sqrt{343}$

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Now, let's apply this concept to the given problem: $\sqrt{343}$. To simplify this square root, we need to find the largest perfect square that divides 343.

Finding the Largest Perfect Square

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To find the largest perfect square that divides 343, we can start by finding the prime factorization of 343. The prime factorization of 343 is $7 \times 7 \times 7$. Since 7 is a perfect square (because it can be expressed as $7 \times 7$), we can rewrite 343 as $7^3$.

Simplifying the Square Root

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Now that we have the prime factorization of 343, we can simplify the square root. We can rewrite $\sqrt{343}$ as $\sqrt{7^3}$. Using the property of square roots that $\sqrt{a^2} = a$, we can simplify this expression to $7\sqrt{7}$.

Conclusion

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In conclusion, the correct answer is $\boxed{7\sqrt{7}}$. This is because we simplified the square root of 343 by finding the largest perfect square that divides 343, which is $7^2$, and then rewriting the expression as $7\sqrt{7}$.

Why is $7\sqrt{7}$ the Correct Answer?

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$7\sqrt{7}$ is the correct answer because it is the simplest form of the square root of 343. When we simplify a square root, we are looking for the largest perfect square that divides the number inside the square root. In this case, the largest perfect square that divides 343 is $7^2$, which is 49. Therefore, we can rewrite $\sqrt{343}$ as $\sqrt{49 \times 7}$, which simplifies to $7\sqrt{7}$.

Why are the Other Options Incorrect?

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The other options are incorrect because they do not simplify the square root of 343 to its simplest form. Option A, $7 \sqrt{49}$, is incorrect because it does not simplify the square root of 343. Option B, $7 \sqrt{7}$, is correct, but option C, 7, is incorrect because it does not take into account the square root. Option D, $49 \sqrt{7}$, is incorrect because it does not simplify the square root of 343.

Final Answer

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The final answer is $\boxed{7\sqrt{7}}$.

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Frequently Asked Questions

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Q: What is the largest perfect square that divides 343?

A: The largest perfect square that divides 343 is $7^2$, which is 49.

Q: How do I simplify the square root of 343?

A: To simplify the square root of 343, you need to find the largest perfect square that divides 343. In this case, the largest perfect square that divides 343 is $7^2$, which is 49. Therefore, you can rewrite $\sqrt{343}$ as $\sqrt{49 \times 7}$, which simplifies to $7\sqrt{7}$.

Q: Why is $7\sqrt{7}$ the correct answer?

A: $7\sqrt{7}$ is the correct answer because it is the simplest form of the square root of 343. When you simplify a square root, you are looking for the largest perfect square that divides the number inside the square root. In this case, the largest perfect square that divides 343 is $7^2$, which is 49. Therefore, you can rewrite $\sqrt{343}$ as $\sqrt{49 \times 7}$, which simplifies to $7\sqrt{7}$.

Q: What is the difference between a perfect square and a non-perfect square?

A: A perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 is a perfect square because it can be expressed as $4 \times 4$, while 17 is not a perfect square because it cannot be expressed as the product of an integer with itself.

Q: How do I find the prime factorization of a number?

A: To find the prime factorization of a number, you need to break it down into its prime factors. For example, the prime factorization of 343 is $7 \times 7 \times 7$.

Q: Why is it important to simplify square roots?

A: Simplifying square roots is important because it helps you to express the square root in its simplest form. This can make it easier to work with the square root in mathematical expressions and equations.

Q: Can you give an example of a square root that is not in its simplest form?

A: Yes, an example of a square root that is not in its simplest form is $\sqrt{18}$. This can be simplified to $3\sqrt{2}$.

Q: How do I know when a square root is in its simplest form?

A: You know when a square root is in its simplest form when it cannot be simplified further. For example, $\sqrt{16}$ is in its simplest form because it cannot be simplified further, while $\sqrt{18}$ is not in its simplest form because it can be simplified to $3\sqrt{2}$.

Conclusion

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In conclusion, simplifying square roots is an important concept in mathematics that can help you to express square roots in their simplest form. By understanding the concept of perfect squares and prime factorization, you can simplify square roots and make it easier to work with them in mathematical expressions and equations.

Final Tips

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• Always look for the largest perfect square that divides the number inside the square root.
• Use the property of square roots that $\sqrt{a^2} = a$ to simplify the square root.
• Check if the square root can be simplified further by looking for any remaining perfect squares.
• Practice simplifying square roots to become more comfortable with the concept.