The Factor Tree For 3,025 Is Shown. What Is The Simplest Form Of 3 , 025 \sqrt{3,025} 3 , 025 ​ ?A. 16 B. 55 C. 5 2 ( 11 2 5^2(11^2 5 2 ( 1 1 2 ] D. 5 2 + ( 11 2 5^2 + (11^2 5 2 + ( 1 1 2 ]

Understanding the Factor Tree for 3025

A factor tree is a visual representation of the factors of a given number. It is a useful tool for breaking down complex numbers into their prime factors. The factor tree for 3025 is shown below:

  3025
 /  \
5   11
|   |
5   11

Prime Factorization of 3025

From the factor tree, we can see that 3025 can be expressed as the product of its prime factors: $5^2 \times 11^2$. This is the prime factorization of 3025.

Simplifying Square Roots

When simplifying square roots, we look for perfect square factors within the radicand (the number under the square root sign). In this case, we have $5^2$ and $11^2$ as perfect square factors.

Applying the Properties of Square Roots

We know that the square root of a product is equal to the product of the square roots. Therefore, we can rewrite $\sqrt{3025}$ as $\sqrt{5^2 \times 11^2}$.

Simplifying the Expression

Using the properties of square roots, we can simplify the expression as follows:

$$\sqrt{5^2 \times 11^2} = \sqrt{5^2} \times \sqrt{11^2} = 5 \times 11 = 55$$

Conclusion

Therefore, the simplest form of $\sqrt{3025}$ is $55$.

Comparing with the Options

Let's compare our answer with the options provided:

A. 16 - This is not the correct answer. B. 55 - This is the correct answer. C. $5^2(11^2)$ - This is the prime factorization of 3025, but it is not the simplest form of $\sqrt{3025}$. D. $5^2 + (11^2)$ - This is not the correct answer.

Final Answer

The final answer is $\boxed{55}$.

Understanding the Factor Tree for 3025

A factor tree is a visual representation of the factors of a given number. It is a useful tool for breaking down complex numbers into their prime factors. The factor tree for 3025 is shown below:

  3025
 /  \
5   11
|   |
5   11

Q: What is the prime factorization of 3025?

A: The prime factorization of 3025 is $5^2 \times 11^2$.

Q: How do you simplify square roots?

A: When simplifying square roots, we look for perfect square factors within the radicand (the number under the square root sign). In this case, we have $5^2$ and $11^2$ as perfect square factors.

Q: What are the properties of square roots?

A: We know that the square root of a product is equal to the product of the square roots. Therefore, we can rewrite $\sqrt{3025}$ as $\sqrt{5^2 \times 11^2}$.

Q: How do you simplify the expression $\sqrt{5^2 \times 11^2}$?

A: Using the properties of square roots, we can simplify the expression as follows:

$$\sqrt{5^2 \times 11^2} = \sqrt{5^2} \times \sqrt{11^2} = 5 \times 11 = 55$$

Q: What is the simplest form of $\sqrt{3025}$?

A: The simplest form of $\sqrt{3025}$ is $55$.

Q: How do you compare your answer with the options provided?

A: Let's compare our answer with the options provided:

A. 16 - This is not the correct answer. B. 55 - This is the correct answer. C. $5^2(11^2)$ - This is the prime factorization of 3025, but it is not the simplest form of $\sqrt{3025}$. D. $5^2 + (11^2)$ - This is not the correct answer.

Q: What is the final answer?

A: The final answer is $\boxed{55}$.

Frequently Asked Questions

Q: What is a factor tree?

A: A factor tree is a visual representation of the factors of a given number.

Q: How do you use a factor tree to simplify square roots?

A: You look for perfect square factors within the radicand (the number under the square root sign) and simplify the expression using the properties of square roots.

Q: What are the properties of square roots?

A: The square root of a product is equal to the product of the square roots.

Q: How do you simplify the expression $\sqrt{5^2 \times 11^2}$?

A: Using the properties of square roots, we can simplify the expression as follows:

$$\sqrt{5^2 \times 11^2} = \sqrt{5^2} \times \sqrt{11^2} = 5 \times 11 = 55$$

Conclusion

In this article, we have discussed the factor tree for 3025 and simplified the square root of 3025 using the properties of square roots. We have also answered some frequently asked questions related to the topic.