Select The Correct Answer.Which Calculation Correctly Writes $\sqrt{80 X^3 Y^2}$ In Simplest Form Using Prime Factorization?A. $\sqrt{80 X^3 Y^2} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot X \cdot X \cdot X \cdot Y \cdot Y} = 4 X Y

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students to master. In this article, we will explore how to simplify radical expressions using prime factorization, with a focus on the correct calculation for $\sqrt{80 x^3 y^2}$.

Understanding Prime Factorization

Prime factorization is a method of breaking down a number into its prime factors. This involves expressing a number as a product of prime numbers. For example, the prime factorization of 80 is $2^4 \cdot 5$. This means that 80 can be expressed as the product of 2 raised to the power of 4 and 5.

Simplifying Radical Expressions

To simplify a radical expression, we need to express the radicand (the number inside the square root) in terms of its prime factors. We can then take out any pairs of prime factors that are raised to an even power.

The Correct Calculation

Let's consider the expression $\sqrt{80 x^3 y^2}$. To simplify this expression, we need to express the radicand in terms of its prime factors.

$\sqrt{80 x^3 y^2} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot x \cdot x \cdot x \cdot y \cdot y}$

We can see that the radicand can be expressed as the product of 2 raised to the power of 4, 5, x raised to the power of 3, and y raised to the power of 2.

Option A: The Incorrect Calculation

One possible calculation for this expression is:

$\sqrt{80 x^3 y^2} = 4 x y$

However, this calculation is incorrect. Let's examine why.

Breaking Down the Calculation

The calculation $4 x y$ assumes that the prime factors of the radicand can be taken out of the square root. However, this is not the case.

The Correct Calculation

The correct calculation is:

$\sqrt{80 x^3 y^2} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot x \cdot x \cdot x \cdot y \cdot y}$

$= 2^2 \cdot x \cdot y \cdot \sqrt{5 \cdot x \cdot y}$

$= 4 x y \sqrt{5 x y}$

Conclusion

In conclusion, the correct calculation for $\sqrt{80 x^3 y^2}$ in simplest form using prime factorization is $4 x y \sqrt{5 x y}$. This calculation involves expressing the radicand in terms of its prime factors and taking out any pairs of prime factors that are raised to an even power.

Common Mistakes

When simplifying radical expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not expressing the radicand in terms of its prime factors: This is a crucial step in simplifying radical expressions.
• Not taking out pairs of prime factors that are raised to an even power: This can result in an incorrect calculation.
• Not checking the final answer: Always check your final answer to ensure that it is correct.

Tips and Tricks

Here are some tips and tricks for simplifying radical expressions:

• Use prime factorization to express the radicand: This will make it easier to simplify the expression.
• Take out pairs of prime factors that are raised to an even power: This will result in a simpler expression.
• Check your final answer: Always check your final answer to ensure that it is correct.

Practice Problems

Here are some practice problems to help you master the skill of simplifying radical expressions:

• Simplify $\sqrt{16 x^4 y^2}$
• Simplify $\sqrt{9 x^3 y^4}$
• Simplify $\sqrt{25 x^2 y^6}$

Conclusion

Introduction

In our previous article, we explored how to simplify radical expressions using prime factorization. In this article, we will answer some frequently asked questions about simplifying radical expressions.

Q: What is prime factorization?

A: Prime factorization is a method of breaking down a number into its prime factors. This involves expressing a number as a product of prime numbers.

Q: How do I simplify a radical expression using prime factorization?

A: To simplify a radical expression using prime factorization, you need to express the radicand (the number inside the square root) in terms of its prime factors. You can then take out any pairs of prime factors that are raised to an even power.

Q: What is the difference between a prime factor and a composite factor?

A: A prime factor is a prime number that divides a number evenly. A composite factor is a product of prime numbers.

Q: How do I know if a number is a prime factor or a composite factor?

A: To determine if a number is a prime factor or a composite factor, you need to check if it can be divided evenly by any other number. If it can be divided evenly, it is a composite factor. If it cannot be divided evenly, it is a prime factor.

Q: What is the difference between a square root and a cube root?

A: A square root is a root that is raised to the power of 1/2. A cube root is a root that is raised to the power of 1/3.

Q: How do I simplify a radical expression with a cube root?

A: To simplify a radical expression with a cube root, you need to express the radicand in terms of its prime factors. You can then take out any pairs of prime factors that are raised to an even power.

Q: What is the difference between a rational number and an irrational number?

A: A rational number is a number that can be expressed as a ratio of two integers. An irrational number is a number that cannot be expressed as a ratio of two integers.

Q: How do I know if a number is rational or irrational?

A: To determine if a number is rational or irrational, you need to check if it can be expressed as a ratio of two integers. If it can be expressed as a ratio of two integers, it is rational. If it cannot be expressed as a ratio of two integers, it is irrational.

Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be expressed as a ratio of two integers. An imaginary number is a number that cannot be expressed as a ratio of two integers.

Q: How do I know if a number is real or imaginary?

A: To determine if a number is real or imaginary, you need to check if it can be expressed as a ratio of two integers. If it can be expressed as a ratio of two integers, it is real. If it cannot be expressed as a ratio of two integers, it is imaginary.

Q: What is the difference between a positive number and a negative number?

A: A positive number is a number that is greater than zero. A negative number is a number that is less than zero.

Q: How do I know if a number is positive or negative?

A: To determine if a number is positive or negative, you need to check if it is greater than or less than zero.

Conclusion

In conclusion, simplifying radical expressions using prime factorization is a crucial skill for students to master. By following the steps outlined in this article, you can simplify radical expressions with ease. Remember to express the radicand in terms of its prime factors, take out pairs of prime factors that are raised to an even power, and check your final answer. With practice, you'll become a pro at simplifying radical expressions in no time!

Practice Problems

Here are some practice problems to help you master the skill of simplifying radical expressions:

• Simplify $\sqrt{16 x^4 y^2}$
• Simplify $\sqrt{9 x^3 y^4}$
• Simplify $\sqrt{25 x^2 y^6}$

Additional Resources

Here are some additional resources to help you learn more about simplifying radical expressions:

• Online tutorials: Websites such as Khan Academy and Mathway offer online tutorials and practice problems to help you learn more about simplifying radical expressions.
• Textbooks: Textbooks such as "Algebra and Trigonometry" by Michael Sullivan and "College Algebra" by James Stewart offer comprehensive coverage of simplifying radical expressions.
• Practice problems: Websites such as IXL and Math Open Reference offer practice problems to help you master the skill of simplifying radical expressions.