What Is The Simplified Form Of $\sqrt{\frac{2160 X^8}{60 X^2}}$? Assume $x \neq 0$.A. $36 X^3$ B. $36 X^2$ C. $6 X^3$ D. $6 X^2$

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the simplified form of the expression 2160x860x2\sqrt{\frac{2160 x^8}{60 x^2}}, assuming x≠0x \neq 0. We will break down the problem into manageable steps and provide a clear explanation of each step.

Step 1: Factor the Numerator and Denominator

To simplify the expression, we need to factor the numerator and denominator. The numerator is 2160x82160 x^8, and the denominator is 60x260 x^2. We can factor out the greatest common factor (GCF) from each expression.

import math

# Define the numerator and denominator
numerator = 2160 * (x ** 8)
denominator = 60 * (x ** 2)

# Factor out the GCF from the numerator and denominator
gcf_numerator = 60 * (x ** 2)
gcf_denominator = 60 * (x ** 2)

# Simplify the expression
simplified_expression = (numerator / gcf_numerator) / (denominator / gcf_denominator)

Step 2: Simplify the Expression

Now that we have factored out the GCF, we can simplify the expression. We can cancel out the common factors in the numerator and denominator.

# Simplify the expression
simplified_expression = (36 * (x ** 6)) / (1 * (x ** 0))

Step 3: Simplify the Square Root

The expression inside the square root is 36x636 x^6. We can simplify this expression by taking the square root of the numerical coefficient and the variable.

# Simplify the square root
simplified_square_root = 6 * x ** 3

Conclusion

In conclusion, the simplified form of the expression 2160x860x2\sqrt{\frac{2160 x^8}{60 x^2}} is 6x36 x^3. This is because we factored out the GCF from the numerator and denominator, simplified the expression, and took the square root of the numerical coefficient and the variable.

Answer

The correct answer is:

  • A. 36x336 x^3

Explanation

The correct answer is 36x336 x^3 because we simplified the expression by factoring out the GCF, canceling out the common factors, and taking the square root of the numerical coefficient and the variable.

Tips and Tricks

When simplifying radical expressions, it's essential to factor out the GCF from the numerator and denominator. This will help you cancel out the common factors and simplify the expression. Additionally, make sure to take the square root of the numerical coefficient and the variable separately.

Common Mistakes

When simplifying radical expressions, it's easy to make mistakes. Some common mistakes include:

  • Not factoring out the GCF from the numerator and denominator
  • Not canceling out the common factors
  • Not taking the square root of the numerical coefficient and the variable separately

By following these tips and tricks, you can avoid common mistakes and simplify radical expressions with ease.

Real-World Applications

Radical expressions have many real-world applications. For example, they are used in physics to describe the motion of objects, in engineering to design buildings and bridges, and in finance to calculate interest rates.

Conclusion

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In our previous article, we explored the simplified form of the expression 2160x860x2\sqrt{\frac{2160 x^8}{60 x^2}}, assuming x≠0x \neq 0. In this article, we will provide a Q&A guide to help you better understand and simplify radical expressions.

Q: What is a radical expression?

A: A radical expression is an expression that contains a square root or other root. It is denoted by the symbol \sqrt{}.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to follow these steps:

  1. Factor the numerator and denominator.
  2. Simplify the expression by canceling out common factors.
  3. Take the square root of the numerical coefficient and the variable separately.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest factor that divides both the numerator and denominator of an expression.

Q: How do I find the GCF of two numbers?

A: To find the GCF of two numbers, you can use the following steps:

  1. List the factors of each number.
  2. Identify the common factors.
  3. Choose the largest common factor.

Q: What is the difference between a rational expression and a radical expression?

A: A rational expression is an expression that contains a fraction, while a radical expression is an expression that contains a square root or other root.

Q: Can I simplify a radical expression with a negative exponent?

A: Yes, you can simplify a radical expression with a negative exponent by following the same steps as before. However, you need to be careful when simplifying expressions with negative exponents.

Q: How do I simplify a radical expression with a variable in the denominator?

A: To simplify a radical expression with a variable in the denominator, you need to follow these steps:

  1. Factor the numerator and denominator.
  2. Simplify the expression by canceling out common factors.
  3. Take the square root of the numerical coefficient and the variable separately.

Q: Can I simplify a radical expression with a fraction in the numerator or denominator?

A: Yes, you can simplify a radical expression with a fraction in the numerator or denominator by following the same steps as before.

Q: What are some common mistakes to avoid when simplifying radical expressions?

A: Some common mistakes to avoid when simplifying radical expressions include:

  • Not factoring out the GCF from the numerator and denominator
  • Not canceling out common factors
  • Not taking the square root of the numerical coefficient and the variable separately

Q: How do I apply radical expressions in real-world problems?

A: Radical expressions have many real-world applications, including physics, engineering, and finance. You can use radical expressions to describe the motion of objects, design buildings and bridges, and calculate interest rates.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill to master in mathematics. By following the steps outlined in this article, you can simplify radical expressions with ease. Remember to factor out the GCF from the numerator and denominator, cancel out common factors, and take the square root of the numerical coefficient and the variable separately. With practice and patience, you can become proficient in simplifying radical expressions and apply them to real-world problems.

Additional Resources

For more information on simplifying radical expressions, check out the following resources:

  • Khan Academy: Simplifying Radical Expressions
  • Mathway: Simplifying Radical Expressions
  • Wolfram Alpha: Simplifying Radical Expressions

Practice Problems

Try the following practice problems to test your skills in simplifying radical expressions:

  1. Simplify the expression 324x681x2\sqrt{\frac{324 x^6}{81 x^2}}.
  2. Simplify the expression 196x849x4\sqrt{\frac{196 x^8}{49 x^4}}.
  3. Simplify the expression 225x6100x3\sqrt{\frac{225 x^6}{100 x^3}}.

Answer Key

  1. 6x26 x^2
  2. 14x214 x^2
  3. 3x323 x^{\frac{3}{2}}