Which Of The Following Is Equivalent To $\frac{\sqrt[3]{32 X^3 Y^6}}{\sqrt[3]{2 X^9 Y^2}}$, Where $x \geq 0$ And $y \geq 0$?A. $\sqrt[3]{16 X^6 Y^4}$B. $\sqrt[3]{\frac{y^4}{16 X^6}}$C. $\sqrt[3]{\frac{16

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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 process of simplifying radical expressions, with a focus on the given problem: 32x3y632x9y23\frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}}. We will break down the solution into manageable steps, using the properties of exponents and radicals to arrive at the final answer.

Understanding the Problem

The given problem involves simplifying a radical expression, which is a mathematical expression that contains a root or a power of a number. In this case, we are dealing with a cube root, denoted by the symbol 3\sqrt[3]{}. The expression is 32x3y632x9y23\frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}}, where x≥0x \geq 0 and y≥0y \geq 0. Our goal is to simplify this expression and find an equivalent form.

Step 1: Simplify the Numerator

To simplify the numerator, we can start by factoring the expression inside the cube root. We can write 32x3y632 x^3 y^6 as 25x3y62^5 x^3 y^6. This allows us to simplify the expression as follows:

32x3y63=25x3y63=222x3y63\sqrt[3]{32 x^3 y^6} = \sqrt[3]{2^5 x^3 y^6} = 2\sqrt[3]{2^2 x^3 y^6}

Step 2: Simplify the Denominator

Next, we can simplify the denominator by factoring the expression inside the cube root. We can write 2x9y22 x^9 y^2 as 2x9y22 x^9 y^2. This allows us to simplify the expression as follows:

2x9y23=2x9y23\sqrt[3]{2 x^9 y^2} = \sqrt[3]{2 x^9 y^2}

Step 3: Simplify the Fraction

Now that we have simplified the numerator and denominator, we can simplify the fraction by dividing the numerator by the denominator. We can write the expression as follows:

32x3y632x9y23=222x3y632x9y23\frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}} = \frac{2\sqrt[3]{2^2 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}}

Step 4: Cancel Out Common Factors

We can simplify the expression further by canceling out common factors between the numerator and denominator. We can write the expression as follows:

222x3y632x9y23=222x3y6x9y23\frac{2\sqrt[3]{2^2 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}} = 2\sqrt[3]{\frac{2^2 x^3 y^6}{x^9 y^2}}

Step 5: Simplify the Expression Inside the Cube Root

We can simplify the expression inside the cube root by canceling out common factors. We can write the expression as follows:

222x3y6x9y23=222x3y4x9y232\sqrt[3]{\frac{2^2 x^3 y^6}{x^9 y^2}} = 2\sqrt[3]{\frac{2^2 x^3 y^4}{x^9 y^2}}

Step 6: Simplify the Expression Further

We can simplify the expression further by canceling out common factors. We can write the expression as follows:

222x3y4x9y23=222y4x632\sqrt[3]{\frac{2^2 x^3 y^4}{x^9 y^2}} = 2\sqrt[3]{\frac{2^2 y^4}{x^6}}

Step 7: Simplify the Expression Inside the Cube Root

We can simplify the expression inside the cube root by canceling out common factors. We can write the expression as follows:

222y4x63=24y416x632\sqrt[3]{\frac{2^2 y^4}{x^6}} = 2\sqrt[3]{\frac{4 y^4}{16 x^6}}

Step 8: Simplify the Expression Further

We can simplify the expression further by canceling out common factors. We can write the expression as follows:

24y416x63=2y416x632\sqrt[3]{\frac{4 y^4}{16 x^6}} = 2\sqrt[3]{\frac{y^4}{16 x^6}}

Conclusion

In conclusion, we have simplified the given radical expression 32x3y632x9y23\frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}} to its equivalent form y416x63\sqrt[3]{\frac{y^4}{16 x^6}}. This was achieved by simplifying the numerator and denominator, canceling out common factors, and simplifying the expression inside the cube root.

Answer

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a root or a power of a number. In this case, we are dealing with a cube root, denoted by the symbol 3\sqrt[3]{}.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can start by factoring the expression inside the cube root. You can then simplify the expression by canceling out common factors between the numerator and denominator.

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

A: A cube root is a root that is raised to the power of 3, while a square root is a root that is raised to the power of 2. In other words, a cube root is denoted by the symbol 3\sqrt[3]{}, while a square root is denoted by the symbol \sqrt{}.

Q: How do I simplify an expression with multiple cube roots?

A: To simplify an expression with multiple cube roots, you can start by simplifying each cube root individually. You can then combine the simplified expressions to arrive at the final answer.

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

A: Yes, you can simplify a radical expression with negative numbers. However, you must remember to follow the rules of exponents and radicals when simplifying the expression.

Q: How do I know when to simplify a radical expression?

A: You should simplify a radical expression when it is necessary to do so in order to arrive at the final answer. Simplifying a radical expression can help to make the expression easier to work with and can also help to avoid errors.

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 the expression inside the cube root
  • Not canceling out common factors between the numerator and denominator
  • Not following the rules of exponents and radicals
  • Not simplifying the expression inside the cube root

Q: How can I practice simplifying radical expressions?

A: You can practice simplifying radical expressions by working through examples and exercises. You can also try simplifying radical expressions on your own and then checking your work to see if you made any mistakes.

Q: What are some real-world applications of simplifying radical expressions?

A: Simplifying radical expressions has many real-world applications, including:

  • Calculating distances and heights in geometry and trigonometry
  • Solving equations and inequalities in algebra and calculus
  • Working with electrical circuits and electronics
  • Calculating areas and volumes in engineering and architecture

Conclusion

In conclusion, simplifying radical expressions is an important skill to master in mathematics. By following the steps outlined in this article, you can simplify radical expressions and arrive at the final answer. Remember to practice simplifying radical expressions regularly to build your skills and confidence.