Simplify 216 \sqrt{216} 216 ​ .A) 4 B) 6 6 6 \sqrt{6} 6 6 ​ C) 7 6 7 \sqrt{6} 7 6 ​ D) 10 2 10 \sqrt{2} 10 2 ​

Introduction

Simplifying square roots is an essential skill in mathematics, particularly in algebra and geometry. It involves expressing a square root in its simplest form, which can be a whole number or a product of a whole number and a square root. In this article, we will focus on simplifying the square root of 216, which is a common problem in mathematics.

What is a Square Root?

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 can be represented by the symbol √, and it is often denoted as the nth root of a number, where n is an integer.

Simplifying Square Roots

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 × 4.

Step 1: Factorize the Number

To simplify the square root of 216, we need to factorize the number 216. We can start by dividing 216 by the smallest perfect square, which is 1. However, we can also try dividing 216 by other perfect squares, such as 4, 9, 16, and so on.

Step 2: Identify the Largest Perfect Square

After factorizing 216, we find that it can be expressed as 2 × 2 × 3 × 3 × 3. We can see that 2 × 2 is a perfect square, and 3 × 3 is also a perfect square. Therefore, the largest perfect square that divides 216 is 2 × 2 × 3 × 3, which equals 36.

Step 3: Simplify the Square Root

Now that we have identified the largest perfect square that divides 216, we can simplify the square root of 216. We can write the square root of 216 as the square root of 36 multiplied by the square root of the remaining factor, which is 3.

The Final Answer

Using the steps above, we can simplify the square root of 216 as follows:

√216 = √(36 × 6) = √36 × √6 = 6√6

Therefore, the correct answer is B) $6 \sqrt{6}$.

Conclusion

Simplifying square roots is an essential skill in mathematics, and it involves expressing a square root in its simplest form. By following the steps outlined in this article, we can simplify the square root of 216 and arrive at the correct answer. We hope that this article has provided a clear and concise guide to simplifying square roots.

Common Mistakes to Avoid

When simplifying square roots, there are several common mistakes to avoid. These include:

• Not identifying the largest perfect square: Failing to identify the largest perfect square that divides the number inside the square root can lead to incorrect simplifications.
• Not simplifying the square root of the remaining factor: Failing to simplify the square root of the remaining factor can lead to incorrect answers.
• Not checking the answer: Failing to check the answer can lead to incorrect simplifications.

Tips and Tricks

When simplifying square roots, there are several tips and tricks to keep in mind. These include:

• Use factorization: Factorizing the number inside the square root can help identify the largest perfect square that divides the number.
• Use the square root of a perfect square: The square root of a perfect square is equal to the number itself.
• Simplify the square root of the remaining factor: Simplifying the square root of the remaining factor can help arrive at the correct answer.

Practice Problems

To practice simplifying square roots, try the following problems:

• Simplify the square root of 144.
• Simplify the square root of 225.
• Simplify the square root of 324.

Answer Key

The answers to the practice problems are:

• √144 = 12
• √225 = 15
• √324 = 18
  Simplifying Square Roots: A Q&A Guide =====================================

Introduction

Simplifying square roots is an essential skill in mathematics, particularly in algebra and geometry. In our previous article, we provided a step-by-step guide on how to simplify the square root of 216. In this article, we will answer some frequently asked questions about simplifying square roots.

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

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 × 4. A square root, on the other hand, 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.

Q: How do I simplify a square root?

A: To simplify a square root, you need to find the largest perfect square that divides the number inside the square root. You can do this by factorizing the number and identifying the largest perfect square that divides it. Then, you can simplify the square root by writing it as the square root of the perfect square multiplied by the square root of the remaining factor.

Q: What is the largest perfect square that divides 144?

A: The largest perfect square that divides 144 is 36. This is because 144 can be expressed as 36 × 4, and 36 is a perfect square.

Q: How do I simplify the square root of 144?

A: To simplify the square root of 144, you need to find the largest perfect square that divides 144. In this case, the largest perfect square that divides 144 is 36. Therefore, you can simplify the square root of 144 as follows:

√144 = √(36 × 4) = √36 × √4 = 6√4 = 6 × 2 = 12

Q: What is the difference between a rational and an irrational number?

A: A rational number is a number that can be expressed as the ratio of two integers. For example, 3/4 is a rational number. An irrational number, on the other hand, is a number that cannot be expressed as the ratio of two integers. For example, the square root of 2 is an irrational number.

Q: Can I simplify an irrational number?

A: No, you cannot simplify an irrational number. Irrational numbers are numbers that cannot be expressed as the ratio of two integers, and therefore, they cannot be simplified.

Q: How do I know if a number is a perfect square or not?

A: To determine if a number is a perfect square or not, you can try to factorize the number and see if it can be expressed as the product of an integer with itself. If it can, then it is a perfect square.

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

A: Some common mistakes to avoid when simplifying square roots include:

• Not identifying the largest perfect square that divides the number inside the square root.
• Not simplifying the square root of the remaining factor.
• Not checking the answer.

Q: How can I practice simplifying square roots?

A: You can practice simplifying square roots by trying the following problems:

• Simplify the square root of 225.
• Simplify the square root of 324.
• Simplify the square root of 400.

Answer Key

The answers to the practice problems are:

• √225 = 15
• √324 = 18
• √400 = 20

Conclusion

Simplifying square roots is an essential skill in mathematics, and it involves expressing a square root in its simplest form. By following the steps outlined in this article, you can simplify square roots and arrive at the correct answer. We hope that this article has provided a clear and concise guide to simplifying square roots.