Simplify The Expression: 2 45 2 \sqrt{45} 2 45 ​

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Introduction

Simplifying expressions involving square roots is a crucial skill in mathematics, particularly in algebra and geometry. In this article, we will focus on simplifying the expression $2 \sqrt{45}$, which involves finding the prime factorization of the number inside the square root and then simplifying the expression accordingly.

Understanding Square Roots

Before we dive into simplifying the expression, let's quickly 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. We can represent square roots using the symbol $\sqrt{}$, so the square root of 16 can be written as $\sqrt{16}$.

Simplifying Square Roots

To simplify a square root, we need to find the prime factorization of the number inside the square root. Prime factorization involves breaking down a number into its smallest prime factors. For example, the prime factorization of 12 is $2^2 \times 3$, because 12 can be divided by 2 twice, and then by 3.

Simplifying the Expression $2 \sqrt{45}$

Now that we have a basic understanding of square roots and simplifying them, let's apply this knowledge to the expression $2 \sqrt{45}$. To simplify this expression, we need to find the prime factorization of 45.

Prime Factorization of 45

The prime factorization of 45 is $3^2 \times 5$. This is because 45 can be divided by 3 twice, and then by 5.

Simplifying the Expression

Now that we have the prime factorization of 45, we can simplify the expression $2 \sqrt{45}$. We can rewrite the expression as $2 \sqrt{3^2 \times 5}$.

Using the Property of Square Roots

We know that the square root of a product is equal to the product of the square roots. Using this property, we can rewrite the expression as $2 \times 3 \times \sqrt{5}$.

Final Simplification

Therefore, the final simplified expression is $6 \sqrt{5}$.

Conclusion

In this article, we simplified the expression $2 \sqrt{45}$ by finding the prime factorization of 45 and then using the property of square roots to simplify the expression. We learned that the prime factorization of 45 is $3^2 \times 5$, and that the square root of a product is equal to the product of the square roots. By applying these concepts, we were able to simplify the expression to $6 \sqrt{5}$.

Additional Examples

Here are a few more examples of simplifying expressions involving square roots:

• $\sqrt{24}$ can be simplified to $2 \sqrt{6}$.
• $\sqrt{72}$ can be simplified to $6 \sqrt{2}$.
• $\sqrt{108}$ can be simplified to $6 \sqrt{3}$.

Tips and Tricks

Here are a few tips and tricks for simplifying expressions involving square roots:

• Always start by finding the prime factorization of the number inside the square root.
• Use the property of square roots to simplify the expression.
• Look for any common factors that can be simplified.
• Practice, practice, practice! The more you practice simplifying expressions involving square roots, the more comfortable you will become with the process.

Final Thoughts

Simplifying expressions involving square roots is an important skill in mathematics, and it requires a combination of understanding the properties of square roots and being able to apply them to simplify expressions. By following the steps outlined in this article, you should be able to simplify expressions involving square roots with ease. Remember to always start by finding the prime factorization of the number inside the square root, and then use the property of square roots to simplify the expression. With practice and patience, you will become a pro at simplifying expressions involving square roots!

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Introduction

In our previous article, we simplified the expression $2 \sqrt{45}$ by finding the prime factorization of 45 and then using the property of square roots to simplify the expression. We learned that the prime factorization of 45 is $3^2 \times 5$, and that the square root of a product is equal to the product of the square roots. By applying these concepts, we were able to simplify the expression to $6 \sqrt{5}$.

Q&A

Here are some frequently asked questions about simplifying expressions involving square roots, along with their answers:

Q: What is the prime factorization of 45?

A: The prime factorization of 45 is $3^2 \times 5$.

Q: How do I simplify the expression $\sqrt{24}$?

A: To simplify the expression $\sqrt{24}$, we need to find the prime factorization of 24. The prime factorization of 24 is $2^3 \times 3$. We can then rewrite the expression as $\sqrt{2^3 \times 3}$, which simplifies to $2 \sqrt{3}$.

Q: How do I simplify the expression $\sqrt{72}$?

A: To simplify the expression $\sqrt{72}$, we need to find the prime factorization of 72. The prime factorization of 72 is $2^3 \times 3^2$. We can then rewrite the expression as $\sqrt{2^3 \times 3^2}$, which simplifies to $6 \sqrt{2}$.

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

A: To simplify the expression $\sqrt{108}$, we need to find the prime factorization of 108. The prime factorization of 108 is $2^2 \times 3^3$. We can then rewrite the expression as $\sqrt{2^2 \times 3^3}$, which simplifies to $6 \sqrt{3}$.

Q: What is the property of square roots that I need to use to simplify expressions?

A: The property of square roots that you need to use to simplify expressions is that the square root of a product is equal to the product of the square roots. This means that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$.

Q: How do I know when to simplify an expression involving a square root?

A: You should simplify an expression involving a square root whenever possible. This means that you should always try to find the prime factorization of the number inside the square root and then use the property of square roots to simplify the expression.

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

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

• Not finding the prime factorization of the number inside the square root.
• Not using the property of square roots to simplify the expression.
• Not looking for any common factors that can be simplified.
• Not practicing, practicing, practicing!

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions involving square roots. We learned that the prime factorization of 45 is $3^2 \times 5$, and that the square root of a product is equal to the product of the square roots. We also learned how to simplify expressions involving square roots, and some common mistakes to avoid when doing so. By following the steps outlined in this article, you should be able to simplify expressions involving square roots with ease.

Additional Resources

Here are some additional resources that you may find helpful when learning about simplifying expressions involving square roots:

• Khan Academy: Simplifying Square Roots
• Mathway: Simplifying Square Roots
• Purplemath: Simplifying Square Roots

Final Thoughts

Simplifying expressions involving square roots is an important skill in mathematics, and it requires a combination of understanding the properties of square roots and being able to apply them to simplify expressions. By following the steps outlined in this article, you should be able to simplify expressions involving square roots with ease. Remember to always start by finding the prime factorization of the number inside the square root, and then use the property of square roots to simplify the expression. With practice and patience, you will become a pro at simplifying expressions involving square roots!