What Is The Simplified Form For $\sqrt{12}$?A. $3 \sqrt{2}$ B. $6 \sqrt{2}$ C. $6 \sqrt{3}$ D. $2 \sqrt{3}$

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Introduction

Simplifying square roots is an essential skill in mathematics, particularly in algebra and geometry. It involves expressing a given number or expression in its simplest radical form, which can be done by factoring the number under the square root sign into its prime factors. In this article, we will explore the simplified form of $\sqrt{12}$ and provide a step-by-step guide on how to simplify it.

Understanding Square Roots

Before we dive into simplifying $\sqrt{12}$, let's briefly review what square roots are. 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. The square root of a number is denoted by the symbol $\sqrt{}$. In this case, we are dealing with the square root of 12, denoted as $\sqrt{12}$.

Simplifying $\sqrt{12}$

To simplify $\sqrt{12}$, we need to factor 12 into its prime factors. The prime factorization of 12 is $2^2 \times 3$. Now, we can rewrite $\sqrt{12}$ as $\sqrt{2^2 \times 3}$. Using the property of square roots that allows us to separate the square root of a product into the product of the square roots, we can rewrite $\sqrt{2^2 \times 3}$ as $\sqrt{2^2} \times \sqrt{3}$.

Applying the Property of Square Roots

Now, let's apply the property of square roots that allows us to simplify the square root of a perfect square. A perfect square is a number that can be expressed as the square of an integer. In this case, $2^2$ is a perfect square, because it can be expressed as the square of 2. Therefore, we can simplify $\sqrt{2^2}$ as 2.

Simplifying the Expression

Now that we have simplified $\sqrt{2^2}$ as 2, we can rewrite the expression $\sqrt{2^2} \times \sqrt{3}$ as $2 \times \sqrt{3}$. This is the simplified form of $\sqrt{12}$.

Conclusion

In conclusion, the simplified form of $\sqrt{12}$ is $2 \times \sqrt{3}$, which can be further simplified as $2\sqrt{3}$. This is the correct answer among the options provided.

Final Answer

The final answer is: $\boxed{2\sqrt{3}}$

Discussion

The simplified form of $\sqrt{12}$ is $2\sqrt{3}$. This can be verified by multiplying $2\sqrt{3}$ by itself, which gives us $12$. Therefore, $2\sqrt{3}$ is indeed the square root of 12.

Related Topics

• Simplifying square roots
• Prime factorization
• Properties of square roots

Example Problems

• Simplify $\sqrt{18}$.
• Simplify $\sqrt{24}$.
• Simplify $\sqrt{36}$.

Solutions

• $\sqrt{18}$ can be simplified as $3\sqrt{2}$.
• $\sqrt{24}$ can be simplified as $2\sqrt{6}$.
• $\sqrt{36}$ can be simplified as $6$.

Practice Problems

• Simplify $\sqrt{48}$.
• Simplify $\sqrt{72}$.
• Simplify $\sqrt{100}$.

Solutions

• $\sqrt{48}$ can be simplified as $4\sqrt{3}$.
• $\sqrt{72}$ can be simplified as $6\sqrt{2}$.
• $\sqrt{100}$ can be simplified as $10$.

Tips and Tricks

• When simplifying square roots, always factor the number under the square root sign into its prime factors.
• Use the property of square roots that allows us to separate the square root of a product into the product of the square roots.
• Simplify the square root of a perfect square by taking the square root of the integer.

Common Mistakes

• Failing to factor the number under the square root sign into its prime factors.
• Not using the property of square roots that allows us to separate the square root of a product into the product of the square roots.
• Not simplifying the square root of a perfect square by taking the square root of the integer.

Conclusion

Simplifying square roots is an essential skill in mathematics, particularly in algebra and geometry. By following the steps outlined in this article, you can simplify $\sqrt{12}$ and other square roots. Remember to factor the number under the square root sign into its prime factors, use the property of square roots that allows us to separate the square root of a product into the product of the square roots, and simplify the square root of a perfect square by taking the square root of the integer. With practice and patience, you can become proficient in simplifying square roots and solving problems involving square roots.

Introduction

Simplifying square roots is an essential skill in mathematics, particularly in algebra and geometry. In our previous article, we explored the simplified form of $\sqrt{12}$ and provided a step-by-step guide on how to simplify it. In this article, we will answer some frequently asked questions about simplifying square roots.

Q: What is the simplified form of $\sqrt{20}$?

A: To simplify $\sqrt{20}$, we need to factor 20 into its prime factors. The prime factorization of 20 is $2^2 \times 5$. Now, we can rewrite $\sqrt{20}$ as $\sqrt{2^2 \times 5}$. Using the property of square roots that allows us to separate the square root of a product into the product of the square roots, we can rewrite $\sqrt{2^2 \times 5}$ as $\sqrt{2^2} \times \sqrt{5}$. Simplifying the square root of $2^2$ as 2, we get $2 \times \sqrt{5}$, which is the simplified form of $\sqrt{20}$.

Q: How do I simplify $\sqrt{48}$?

A: To simplify $\sqrt{48}$, we need to factor 48 into its prime factors. The prime factorization of 48 is $2^4 \times 3$. Now, we can rewrite $\sqrt{48}$ as $\sqrt{2^4 \times 3}$. Using the property of square roots that allows us to separate the square root of a product into the product of the square roots, we can rewrite $\sqrt{2^4 \times 3}$ as $\sqrt{2^4} \times \sqrt{3}$. Simplifying the square root of $2^4$ as 4, we get $4 \times \sqrt{3}$, which is the simplified form of $\sqrt{48}$.

Q: What is the simplified form of $\sqrt{75}$?

A: To simplify $\sqrt{75}$, we need to factor 75 into its prime factors. The prime factorization of 75 is $3^2 \times 5$. Now, we can rewrite $\sqrt{75}$ as $\sqrt{3^2 \times 5}$. Using the property of square roots that allows us to separate the square root of a product into the product of the square roots, we can rewrite $\sqrt{3^2 \times 5}$ as $\sqrt{3^2} \times \sqrt{5}$. Simplifying the square root of $3^2$ as 3, we get $3 \times \sqrt{5}$, which is the simplified form of $\sqrt{75}$.

Q: How do I simplify $\sqrt{108}$?

A: To simplify $\sqrt{108}$, we need to factor 108 into its prime factors. The prime factorization of 108 is $2^2 \times 3^3$. Now, we can rewrite $\sqrt{108}$ as $\sqrt{2^2 \times 3^3}$. Using the property of square roots that allows us to separate the square root of a product into the product of the square roots, we can rewrite $\sqrt{2^2 \times 3^3}$ as $\sqrt{2^2} \times \sqrt{3^3}$. Simplifying the square root of $2^2$ as 2 and the square root of $3^3$ as 3$\sqrt{3}$, we get $2 \times 3\sqrt{3}$, which is the simplified form of $\sqrt{108}$.

Q: What is the simplified form of $\sqrt{144}$?

A: To simplify $\sqrt{144}$, we need to factor 144 into its prime factors. The prime factorization of 144 is $2^4 \times 3^2$. Now, we can rewrite $\sqrt{144}$ as $\sqrt{2^4 \times 3^2}$. Using the property of square roots that allows us to separate the square root of a product into the product of the square roots, we can rewrite $\sqrt{2^4 \times 3^2}$ as $\sqrt{2^4} \times \sqrt{3^2}$. Simplifying the square root of $2^4$ as 4 and the square root of $3^2$ as 3, we get $4 \times 3$, which is the simplified form of $\sqrt{144}$.

Q: How do I simplify $\sqrt{225}$?

A: To simplify $\sqrt{225}$, we need to factor 225 into its prime factors. The prime factorization of 225 is $3^2 \times 5^2$. Now, we can rewrite $\sqrt{225}$ as $\sqrt{3^2 \times 5^2}$. Using the property of square roots that allows us to separate the square root of a product into the product of the square roots, we can rewrite $\sqrt{3^2 \times 5^2}$ as $\sqrt{3^2} \times \sqrt{5^2}$. Simplifying the square root of $3^2$ as 3 and the square root of $5^2$ as 5, we get $3 \times 5$, which is the simplified form of $\sqrt{225}$.

Conclusion

Simplifying square roots is an essential skill in mathematics, particularly in algebra and geometry. By following the steps outlined in this article, you can simplify square roots and solve problems involving square roots. Remember to factor the number under the square root sign into its prime factors, use the property of square roots that allows us to separate the square root of a product into the product of the square roots, and simplify the square root of a perfect square by taking the square root of the integer. With practice and patience, you can become proficient in simplifying square roots and solving problems involving square roots.

Final Tips

• Always factor the number under the square root sign into its prime factors.
• Use the property of square roots that allows us to separate the square root of a product into the product of the square roots.
• Simplify the square root of a perfect square by taking the square root of the integer.
• Practice and patience are key to becoming proficient in simplifying square roots and solving problems involving square roots.

Related Topics

• Simplifying square roots
• Prime factorization
• Properties of square roots

Example Problems

• Simplify $\sqrt{30}$.
• Simplify $\sqrt{50}$.
• Simplify $\sqrt{90}$.

Solutions

• $\sqrt{30}$ can be simplified as $3\sqrt{10}$.
• $\sqrt{50}$ can be simplified as $5\sqrt{2}$.
• $\sqrt{90}$ can be simplified as $3\sqrt{10}$.

Practice Problems

• Simplify $\sqrt{60}$.
• Simplify $\sqrt{80}$.
• Simplify $\sqrt{120}$.

Solutions

• $\sqrt{60}$ can be simplified as $2\sqrt{15}$.
• $\sqrt{80}$ can be simplified as $4\sqrt{5}$.
• $\sqrt{120}$ can be simplified as $2\sqrt{30}$.

Conclusion

Simplifying square roots is an essential skill in mathematics, particularly in algebra and geometry. By following the steps outlined in this article, you can simplify square roots and solve problems involving square roots. Remember to factor the number under the square root sign into its prime factors, use the property of square roots that allows us to separate the square root of a product into the product of the square roots, and simplify the square root of a perfect square by taking the square root of the integer. With practice and patience, you can become proficient in simplifying square roots and solving problems involving square roots.