What Is The Simplified Form Of 400 X 100 \sqrt{400 X^{100}} 400 X 100 ​ ?A. 200 X 10 200 X^{10} 200 X 10 B. 200 X 50 200 X^{50} 200 X 50 C. 20 X 10 20 X^{10} 20 X 10 D. 20 X 50 20 X^{50} 20 X 50

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Understanding the Problem

The problem requires us to simplify the expression $\sqrt{400 x^{100}}$. To do this, we need to apply the properties of radicals and exponents. The expression inside the square root can be broken down into two separate components: a numerical part and an exponential part.

Breaking Down the Expression

The numerical part is 400, which can be expressed as $20^2$. The exponential part is $x^{100}$, which can be expressed as $(x^{10})^2$. By applying the property of radicals that states $\sqrt{a^2} = a$, we can simplify the expression.

Simplifying the Numerical Part

The numerical part, $20^2$, can be simplified using the property of radicals. Since $20^2$ is a perfect square, we can take the square root of it, which is 20.

Simplifying the Exponential Part

The exponential part, $(x^{10})^2$, can also be simplified using the property of radicals. Since $(x^{10})^2$ is a perfect square, we can take the square root of it, which is $x^{10}$.

Combining the Simplified Parts

Now that we have simplified the numerical and exponential parts, we can combine them to get the final simplified form of the expression. The simplified form is $20x^{10}$.

Comparing with the Options

Let's compare the simplified form with the options provided:

• A. $200 x^{10}$: This option is incorrect because the coefficient is 200, not 20.
• B. $200 x^{50}$: This option is incorrect because the coefficient is 200, not 20, and the exponent is 50, not 10.
• C. $20 x^{10}$: This option is correct because it matches the simplified form we obtained.
• D. $20 x^{50}$: This option is incorrect because the exponent is 50, not 10.

Conclusion

In conclusion, the simplified form of $\sqrt{400 x^{100}}$ is $20x^{10}$. This can be obtained by breaking down the expression into its numerical and exponential parts, simplifying each part using the properties of radicals and exponents, and then combining the simplified parts.

Final Answer

The final answer is $\boxed{20x^{10}}$.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Break down the expression into its numerical and exponential parts: $\sqrt{400 x^{100}} = \sqrt{20^2 (x^{10})^2}$.
2. Simplify the numerical part using the property of radicals: $\sqrt{20^2} = 20$.
3. Simplify the exponential part using the property of radicals: $\sqrt{(x^{10})^2} = x^{10}$.
4. Combine the simplified parts: $20x^{10}$.

Common Mistakes

Here are some common mistakes to avoid when simplifying the expression:

• Not breaking down the expression into its numerical and exponential parts.
• Not simplifying the numerical part using the property of radicals.
• Not simplifying the exponential part using the property of radicals.
• Not combining the simplified parts correctly.

Tips and Tricks

Here are some tips and tricks to help you simplify the expression:

• Make sure to break down the expression into its numerical and exponential parts.
• Use the property of radicals to simplify the numerical and exponential parts.
• Combine the simplified parts correctly.
• Check your answer by plugging it back into the original expression.

Real-World Applications

Here are some real-world applications of simplifying the expression:

• Simplifying expressions is an important skill in mathematics and science.
• It can be used to solve problems in algebra, geometry, and calculus.
• It can also be used to simplify complex expressions in physics and engineering.

Practice Problems

Here are some practice problems to help you practice simplifying expressions:

• Simplify the expression $\sqrt{36 x^8}$.
• Simplify the expression $\sqrt{49 x^{12}}$.
• Simplify the expression $\sqrt{25 x^{20}}$.

Conclusion

In conclusion, simplifying the expression $\sqrt{400 x^{100}}$ requires breaking down the expression into its numerical and exponential parts, simplifying each part using the properties of radicals and exponents, and then combining the simplified parts. The final simplified form is $20x^{10}$.

Introduction

Simplifying expressions with radicals and exponents is an important skill in mathematics and science. In this article, we will answer some common questions and provide examples to help you understand how to simplify expressions with radicals and exponents.

Q: What is the difference between a radical and an exponent?

A: A radical is a symbol that represents the square root of a number, while an exponent is a number that represents the power to which a base number is raised.

Q: How do I simplify an expression with a radical?

A: To simplify an expression with a radical, you need to break down the expression into its numerical and exponential parts, simplify each part using the properties of radicals and exponents, and then combine the simplified parts.

Q: What is the property of radicals that states $\sqrt{a^2} = a$?

A: This property states that the square root of a perfect square is equal to the number itself. For example, $\sqrt{16} = 4$ because $4^2 = 16$.

Q: How do I simplify an expression with an exponent?

A: To simplify an expression with an exponent, you need to break down the expression into its base and exponent parts, simplify each part using the properties of exponents, and then combine the simplified parts.

Q: What is the property of exponents that states $a^{m+n} = a^m \cdot a^n$?

A: This property states that when you multiply two numbers with the same base, you can add their exponents. For example, $2^3 \cdot 2^4 = 2^{3+4} = 2^7$.

Q: How do I simplify an expression with both radicals and exponents?

A: To simplify an expression with both radicals and exponents, you need to break down the expression into its numerical, exponential, and radical parts, simplify each part using the properties of radicals and exponents, and then combine the simplified parts.

Q: What is the final simplified form of $\sqrt{400 x^{100}}$?

A: The final simplified form of $\sqrt{400 x^{100}}$ is $20x^{10}$.

Q: What are some common mistakes to avoid when simplifying expressions with radicals and exponents?

A: Some common mistakes to avoid include not breaking down the expression into its numerical and exponential parts, not simplifying the numerical and exponential parts using the properties of radicals and exponents, and not combining the simplified parts correctly.

Q: What are some real-world applications of simplifying expressions with radicals and exponents?

A: Simplifying expressions with radicals and exponents is an important skill in mathematics and science. It can be used to solve problems in algebra, geometry, and calculus, and can also be used to simplify complex expressions in physics and engineering.

Q: How can I practice simplifying expressions with radicals and exponents?

A: You can practice simplifying expressions with radicals and exponents by working through practice problems, such as simplifying the expression $\sqrt{36 x^8}$ or $\sqrt{49 x^{12}}$.

Conclusion

In conclusion, simplifying expressions with radicals and exponents requires breaking down the expression into its numerical and exponential parts, simplifying each part using the properties of radicals and exponents, and then combining the simplified parts. By following these steps and practicing with examples, you can become proficient in simplifying expressions with radicals and exponents.

Final Answer

The final answer is $\boxed{20x^{10}}$.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Break down the expression into its numerical and exponential parts: $\sqrt{400 x^{100}} = \sqrt{20^2 (x^{10})^2}$.
2. Simplify the numerical part using the property of radicals: $\sqrt{20^2} = 20$.
3. Simplify the exponential part using the property of radicals: $\sqrt{(x^{10})^2} = x^{10}$.
4. Combine the simplified parts: $20x^{10}$.

Common Mistakes

Here are some common mistakes to avoid when simplifying expressions with radicals and exponents:

• Not breaking down the expression into its numerical and exponential parts.
• Not simplifying the numerical and exponential parts using the properties of radicals and exponents.
• Not combining the simplified parts correctly.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions with radicals and exponents:

• Make sure to break down the expression into its numerical and exponential parts.
• Use the property of radicals to simplify the numerical and exponential parts.
• Combine the simplified parts correctly.
• Check your answer by plugging it back into the original expression.

Real-World Applications

Here are some real-world applications of simplifying expressions with radicals and exponents:

• Simplifying expressions is an important skill in mathematics and science.
• It can be used to solve problems in algebra, geometry, and calculus.
• It can also be used to simplify complex expressions in physics and engineering.

Practice Problems

Here are some practice problems to help you practice simplifying expressions with radicals and exponents:

• Simplify the expression $\sqrt{36 x^8}$.
• Simplify the expression $\sqrt{49 x^{12}}$.
• Simplify the expression $\sqrt{25 x^{20}}$.

Conclusion

In conclusion, simplifying expressions with radicals and exponents requires breaking down the expression into its numerical and exponential parts, simplifying each part using the properties of radicals and exponents, and then combining the simplified parts. By following these steps and practicing with examples, you can become proficient in simplifying expressions with radicals and exponents.