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

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Understanding the Problem

When dealing with radical expressions, it's essential to understand the properties of exponents and roots. In this problem, we're given an expression involving cube roots and variables xx and yy. Our goal is to simplify the given expression and determine which of the provided options is equivalent.

The Given Expression

The given expression is 32x3y632x9y23\frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}}. To simplify this expression, we need to apply the properties of exponents and roots.

Applying the Quotient Rule for Radicals

The quotient rule for radicals states that anbn=abn\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}. We can apply this rule to the given expression:

32x3y632x9y23=32x3y62x9y23\frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}} = \sqrt[3]{\frac{32 x^3 y^6}{2 x^9 y^2}}

Simplifying the Expression

Now, let's simplify the expression inside the cube root:

32x3y62x9y2=16x3y4⋅1x9y2\frac{32 x^3 y^6}{2 x^9 y^2} = 16 x^3 y^4 \cdot \frac{1}{x^9 y^2}

Applying the Product Rule for Exponents

The product rule for exponents states that am⋅an=am+na^m \cdot a^n = a^{m+n}. We can apply this rule to simplify the expression:

16x3y4⋅1x9y2=16x3−9y4−2=16x−6y216 x^3 y^4 \cdot \frac{1}{x^9 y^2} = 16 x^{3-9} y^{4-2} = 16 x^{-6} y^2

Simplifying Negative Exponents

A negative exponent indicates that we need to take the reciprocal of the expression. In this case, we have x−6x^{-6}, which is equivalent to 1x6\frac{1}{x^6}:

16x−6y2=16y2x616 x^{-6} y^2 = \frac{16 y^2}{x^6}

Simplifying the Cube Root

Now, let's simplify the cube root:

16y2x63=16x6⋅y23\sqrt[3]{\frac{16 y^2}{x^6}} = \sqrt[3]{\frac{16}{x^6} \cdot y^2}

Applying the Product Rule for Radicals

The product rule for radicals states that an⋅bn=a⋅bn\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}. We can apply this rule to simplify the expression:

16x6⋅y23=16y2x63\sqrt[3]{\frac{16}{x^6} \cdot y^2} = \sqrt[3]{\frac{16 y^2}{x^6}}

Simplifying the Expression

Now, let's simplify the expression:

16y2x63=16x6⋅y23\sqrt[3]{\frac{16 y^2}{x^6}} = \sqrt[3]{\frac{16}{x^6} \cdot y^2}

Comparing with the Options

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

A. 16x6y43\sqrt[3]{16 x^6 y^4} B. y416x63\sqrt[3]{\frac{y^4}{16 x^6}} C. 16y2x63\sqrt[3]{\frac{16 y^2}{x^6}}

Conclusion

Based on the simplification, we can see that the correct option is:

C. 16y2x63\sqrt[3]{\frac{16 y^2}{x^6}}

This option is equivalent to the simplified expression, and it meets the given conditions.

Final Answer

The final answer is C. 16y2x63\sqrt[3]{\frac{16 y^2}{x^6}}.

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Understanding the Problem

When dealing with radical expressions, it's essential to understand the properties of exponents and roots. In this article, we'll provide a step-by-step guide on how to simplify radical expressions and answer some common questions related to this topic.

Q&A: Simplifying Radical Expressions

Q: What is the quotient rule for radicals?

A: The quotient rule for radicals states that anbn=abn\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}. This rule allows us to simplify expressions involving cube roots and other roots.

Q: How do I apply the quotient rule for radicals?

A: To apply the quotient rule for radicals, simply divide the numerator and denominator by the same root. For example, 32x3y632x9y23=32x3y62x9y23\frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}} = \sqrt[3]{\frac{32 x^3 y^6}{2 x^9 y^2}}.

Q: What is the product rule for exponents?

A: The product rule for exponents states that am⋅an=am+na^m \cdot a^n = a^{m+n}. This rule allows us to simplify expressions involving exponents.

Q: How do I apply the product rule for exponents?

A: To apply the product rule for exponents, simply add the exponents of the same base. For example, 16x3y4⋅1x9y2=16x3−9y4−2=16x−6y216 x^3 y^4 \cdot \frac{1}{x^9 y^2} = 16 x^{3-9} y^{4-2} = 16 x^{-6} y^2.

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

A: A positive exponent indicates that we need to multiply the expression by itself as many times as the exponent. A negative exponent indicates that we need to take the reciprocal of the expression.

Q: How do I simplify a negative exponent?

A: To simplify a negative exponent, simply take the reciprocal of the expression. For example, x−6=1x6x^{-6} = \frac{1}{x^6}.

Q: What is the cube root of a fraction?

A: The cube root of a fraction can be simplified by taking the cube root of the numerator and denominator separately. For example, 16y2x63=16x6⋅y23\sqrt[3]{\frac{16 y^2}{x^6}} = \sqrt[3]{\frac{16}{x^6} \cdot y^2}.

Q: How do I compare a simplified expression with the options?

A: To compare a simplified expression with the options, simply look for the option that is equivalent to the simplified expression. In this case, the correct option is C. 16y2x63\sqrt[3]{\frac{16 y^2}{x^6}}.

Common Mistakes to Avoid

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

  • Not applying the quotient rule for radicals
  • Not applying the product rule for exponents
  • Not simplifying negative exponents
  • Not comparing the simplified expression with the options

Conclusion

Simplifying radical expressions can be a challenging task, but with the right approach and practice, it can become easier. By understanding the properties of exponents and roots, you can simplify radical expressions and solve problems with confidence.

Final Tips

Here are some final tips to help you simplify radical expressions:

  • Practice, practice, practice: The more you practice, the more comfortable you'll become with simplifying radical expressions.
  • Use the quotient rule for radicals: The quotient rule for radicals is a powerful tool that can help you simplify expressions involving cube roots and other roots.
  • Simplify negative exponents: Negative exponents can be simplified by taking the reciprocal of the expression.
  • Compare the simplified expression with the options: This will help you determine which option is equivalent to the simplified expression.

By following these tips and practicing regularly, you'll become a pro at simplifying radical expressions in no time!