Which Of The Following Expressions Are Equivalent To 200 \sqrt{200} 200 ​ ?A. 5 8 5 \sqrt{8} 5 8 ​ B. 10 2 10 \sqrt{2} 10 2 ​ C. 20 10 20 \sqrt{10} 20 10 ​ D. 10 20 10 \sqrt{20} 10 20 ​ Show Your Work Here.

Understanding Square Roots

In mathematics, 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. Square roots are denoted by the symbol $\sqrt{}$. In this article, we will explore the concept of simplifying square roots and evaluate the expressions equivalent to $\sqrt{200}$.

Simplifying $\sqrt{200}$

To simplify $\sqrt{200}$, we need to find the largest perfect square that divides 200. 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 multiplied by 4.

The prime factorization of 200 is $2^3 \times 5^2$. We can rewrite $\sqrt{200}$ as $\sqrt{2^3 \times 5^2}$. Since $5^2$ is a perfect square, we can simplify it as $5 \times 5 = 25$. Therefore, $\sqrt{200}$ can be rewritten as $\sqrt{2^3 \times 25}$.

Simplifying Further

We can simplify $\sqrt{2^3 \times 25}$ by breaking it down into two separate square roots: $\sqrt{2^3}$ and $\sqrt{25}$. Since $2^3$ is not a perfect square, we cannot simplify it further. However, $\sqrt{25}$ can be simplified as $5$, because 5 multiplied by 5 equals 25.

Therefore, $\sqrt{200}$ can be simplified as $\sqrt{2^3} \times 5$. We can further simplify $\sqrt{2^3}$ as $\sqrt{2} \times \sqrt{2} \times \sqrt{2}$, which equals $2\sqrt{2}$.

Evaluating the Expressions

Now that we have simplified $\sqrt{200}$ as $2\sqrt{2} \times 5$, we can evaluate the expressions given in the options.

Option A: $5 \sqrt{8}$

To evaluate this expression, we need to simplify $\sqrt{8}$. The prime factorization of 8 is $2^3$. We can rewrite $\sqrt{8}$ as $\sqrt{2^3}$, which equals $2\sqrt{2}$.

Therefore, $5 \sqrt{8}$ can be rewritten as $5 \times 2\sqrt{2}$, which equals $10\sqrt{2}$.

Option B: $10 \sqrt{2}$

This expression is already simplified, and it equals $10\sqrt{2}$.

Option C: $20 \sqrt{10}$

To evaluate this expression, we need to simplify $\sqrt{10}$. The prime factorization of 10 is $2 \times 5$. We can rewrite $\sqrt{10}$ as $\sqrt{2 \times 5}$, which equals $\sqrt{2} \times \sqrt{5}$.

Therefore, $20 \sqrt{10}$ can be rewritten as $20 \times \sqrt{2} \times \sqrt{5}$, which equals $20\sqrt{10}$.

Option D: $10 \sqrt{20}$

To evaluate this expression, we need to simplify $\sqrt{20}$. The prime factorization of 20 is $2^2 \times 5$. We can rewrite $\sqrt{20}$ as $\sqrt{2^2 \times 5}$, which equals $2\sqrt{5}$.

Therefore, $10 \sqrt{20}$ can be rewritten as $10 \times 2\sqrt{5}$, which equals $20\sqrt{5}$.

Conclusion

In conclusion, the expressions equivalent to $\sqrt{200}$ are $5 \sqrt{8}$ and $10 \sqrt{2}$. The other options, $20 \sqrt{10}$ and $10 \sqrt{20}$, are not equivalent to $\sqrt{200}$.

Final Answer

The final answer is:

• Option A: $5 \sqrt{8}$ is equivalent to $\sqrt{200}$.
• Option B: $10 \sqrt{2}$ is equivalent to $\sqrt{200}$.
• Option C: $20 \sqrt{10}$ is not equivalent to $\sqrt{200}$.
• Option D: $10 \sqrt{20}$ is not equivalent to $\sqrt{200}$.
  

Q: What is a square root?

A: 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.

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. 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 multiplied by 4.

Q: What is the prime factorization of a number?

A: The prime factorization of a number is the expression of the number as a product of prime numbers. For example, the prime factorization of 200 is $2^3 \times 5^2$.

Q: How do I simplify $\sqrt{200}$?

A: To simplify $\sqrt{200}$, you need to find the largest perfect square that divides 200. The prime factorization of 200 is $2^3 \times 5^2$. We can rewrite $\sqrt{200}$ as $\sqrt{2^3 \times 5^2}$. Since $5^2$ is a perfect square, we can simplify it as $5 \times 5 = 25$. Therefore, $\sqrt{200}$ can be rewritten as $\sqrt{2^3 \times 25}$.

Q: How do I simplify $\sqrt{2^3 \times 25}$?

A: We can simplify $\sqrt{2^3 \times 25}$ by breaking it down into two separate square roots: $\sqrt{2^3}$ and $\sqrt{25}$. Since $2^3$ is not a perfect square, we cannot simplify it further. However, $\sqrt{25}$ can be simplified as $5$, because 5 multiplied by 5 equals 25.

Q: How do I simplify $\sqrt{2^3}$?

A: We can simplify $\sqrt{2^3}$ as $\sqrt{2} \times \sqrt{2} \times \sqrt{2}$, which equals $2\sqrt{2}$.

Q: What is the final simplified form of $\sqrt{200}$?

A: The final simplified form of $\sqrt{200}$ is $2\sqrt{2} \times 5$, which equals $10\sqrt{2}$.

Q: How do I evaluate the expressions given in the options?

A: To evaluate the expressions given in the options, you need to simplify each expression separately. For example, to evaluate $5 \sqrt{8}$, you need to simplify $\sqrt{8}$ as $2\sqrt{2}$, and then multiply it by 5.

Q: What is the final answer for each option?

A: The final answer for each option is:

• Option A: $5 \sqrt{8}$ is equivalent to $10\sqrt{2}$.
• Option B: $10 \sqrt{2}$ is equivalent to $10\sqrt{2}$.
• Option C: $20 \sqrt{10}$ is not equivalent to $\sqrt{200}$.
• Option D: $10 \sqrt{20}$ is not equivalent to $\sqrt{200}$.

Q: What is the most important thing to remember when simplifying square roots?

A: The most important thing to remember when simplifying square roots is to find the largest perfect square that divides the number. This will help you simplify the square root and find the final answer.