What Is The Simplified Form Of The Following Expression? Assume $y \neq 0$.A. $\sqrt[3]{\frac{12 X^2}{16 Y}}$B. $\frac{2\left(\sqrt[3]{6 X^2 Y^2}\right)}{y}$C. $\frac{\sqrt[3]{12 X^2 Y}}{2 Y}$D.

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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 a given expression, assuming that $y \neq 0$. We will break down the expression into its constituent parts, apply the necessary rules and formulas, and arrive at the simplified form.

Understanding the Expression

The given expression is $\sqrt[3]{\frac{12 x^2}{16 y}}$. To simplify this expression, we need to understand the properties of radical expressions and the rules for simplifying them.

Radical Expressions: A Brief Overview

A radical expression is a mathematical expression that contains a root or a power of a number. The most common radical expressions are square roots, cube roots, and nth roots. The general form of a radical expression is $\sqrt[n]{a}$, where $a$ is the radicand and $n$ is the index of the root.

Simplifying Radical Expressions: Rules and Formulas

To simplify a radical expression, we need to apply the following rules and formulas:

  • Product Rule: $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$
  • Quotient Rule: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$
  • Power Rule: $\sqrt[n]{a^m} = a^{\frac{m}{n}}$

Simplifying the Given Expression

Now that we have a good understanding of radical expressions and the rules for simplifying them, let's apply these rules to simplify the given expression.

12x216y3\sqrt[3]{\frac{12 x^2}{16 y}}

To simplify this expression, we can start by factoring the numerator and denominator.

12x216y3=3โ‹…4โ‹…x24โ‹…4โ‹…y3\sqrt[3]{\frac{12 x^2}{16 y}} = \sqrt[3]{\frac{3 \cdot 4 \cdot x^2}{4 \cdot 4 \cdot y}}

Now, we can cancel out the common factors in the numerator and denominator.

3โ‹…4โ‹…x24โ‹…4โ‹…y3=3x24y3\sqrt[3]{\frac{3 \cdot 4 \cdot x^2}{4 \cdot 4 \cdot y}} = \sqrt[3]{\frac{3 x^2}{4 y}}

Next, we can apply the quotient rule to simplify the expression further.

3x24y3=3x234y3\sqrt[3]{\frac{3 x^2}{4 y}} = \frac{\sqrt[3]{3 x^2}}{\sqrt[3]{4 y}}

Now, we can apply the power rule to simplify the expression even further.

3x234y3=x23y13โ‹…213\frac{\sqrt[3]{3 x^2}}{\sqrt[3]{4 y}} = \frac{x^{\frac{2}{3}}}{y^{\frac{1}{3}} \cdot 2^{\frac{1}{3}}}

Finally, we can simplify the expression by canceling out the common factors.

x23y13โ‹…213=x23213โ‹…y13\frac{x^{\frac{2}{3}}}{y^{\frac{1}{3}} \cdot 2^{\frac{1}{3}}} = \frac{x^{\frac{2}{3}}}{2^{\frac{1}{3}} \cdot y^{\frac{1}{3}}}

Conclusion

In this article, we simplified the given expression $\sqrt[3]{\frac{12 x^2}{16 y}}$, assuming that $y \neq 0$. We applied the necessary rules and formulas to simplify the expression, and arrived at the simplified form $\frac{x{\frac{2}{3}}}{2{\frac{1}{3}} \cdot y^{\frac{1}{3}}}$.

Comparison with Answer Choices

Now that we have simplified the expression, let's compare it with the answer choices.

A. $\sqrt[3]{\frac{12 x^2}{16 y}}$

This is the original expression, and it is not in its simplest form.

B. $\frac{2\left(\sqrt[3]{6 x^2 y^2}\right)}{y}$

This expression is not equivalent to the simplified expression we arrived at.

C. $\frac{\sqrt[3]{12 x^2 y}}{2 y}$

This expression is not equivalent to the simplified expression we arrived at.

D. $\frac{\sqrt[3]{12 x^2 y}}{2 y}$

This expression is not equivalent to the simplified expression we arrived at.

Conclusion

In conclusion, the simplified form of the given expression $\sqrt[3]{\frac{12 x^2}{16 y}}$, assuming that $y \neq 0$, is $\frac{x{\frac{2}{3}}}{2{\frac{1}{3}} \cdot y^{\frac{1}{3}}}$.

Final Answer

Introduction

In our previous article, we explored the simplified form of a given expression, assuming that $y \neq 0$. We applied the necessary rules and formulas to simplify the expression, and arrived at the simplified form $\frac{x{\frac{2}{3}}}{2{\frac{1}{3}} \cdot y^{\frac{1}{3}}}$.

In this article, we will provide a Q&A guide to help you better understand the concept of simplifying radical expressions. We will answer some common questions and provide examples to illustrate the concepts.

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

A: A radical expression is a mathematical expression that contains a root or a power of a number. A simplified radical expression is a radical expression that has been simplified using the necessary rules and formulas.

Q: What are the rules for simplifying radical expressions?

A: The rules for simplifying radical expressions are:

  • Product Rule: $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$
  • Quotient Rule: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$
  • Power Rule: $\sqrt[n]{a^m} = a^{\frac{m}{n}}$

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to apply the necessary rules and formulas. Here are the steps:

  1. Factor the numerator and denominator.
  2. Cancel out the common factors.
  3. Apply the quotient rule.
  4. Apply the power rule.

Q: What is the simplified form of the expression $\sqrt[3]{\frac{24 x^3}{27 y^3}}$?

A: To simplify this expression, we can start by factoring the numerator and denominator.

24x327y33=3โ‹…8โ‹…x33โ‹…3โ‹…3โ‹…y33\sqrt[3]{\frac{24 x^3}{27 y^3}} = \sqrt[3]{\frac{3 \cdot 8 \cdot x^3}{3 \cdot 3 \cdot 3 \cdot y^3}}

Now, we can cancel out the common factors in the numerator and denominator.

3โ‹…8โ‹…x33โ‹…3โ‹…3โ‹…y33=8x333y33\sqrt[3]{\frac{3 \cdot 8 \cdot x^3}{3 \cdot 3 \cdot 3 \cdot y^3}} = \sqrt[3]{\frac{8 x^3}{3^3 y^3}}

Next, we can apply the quotient rule to simplify the expression further.

8x333y33=8x3333y33\sqrt[3]{\frac{8 x^3}{3^3 y^3}} = \frac{\sqrt[3]{8 x^3}}{\sqrt[3]{3^3 y^3}}

Now, we can apply the power rule to simplify the expression even further.

8x3333y33=2x3y\frac{\sqrt[3]{8 x^3}}{\sqrt[3]{3^3 y^3}} = \frac{2 x}{3 y}

Q: What is the simplified form of the expression $\sqrt[4]{\frac{16 x^4}{81 y^4}}$?

A: To simplify this expression, we can start by factoring the numerator and denominator.

16x481y44=2โ‹…2โ‹…2โ‹…2โ‹…x43โ‹…3โ‹…3โ‹…3โ‹…y44\sqrt[4]{\frac{16 x^4}{81 y^4}} = \sqrt[4]{\frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot x^4}{3 \cdot 3 \cdot 3 \cdot 3 \cdot y^4}}

Now, we can cancel out the common factors in the numerator and denominator.

2โ‹…2โ‹…2โ‹…2โ‹…x43โ‹…3โ‹…3โ‹…3โ‹…y44=24x434y44\sqrt[4]{\frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot x^4}{3 \cdot 3 \cdot 3 \cdot 3 \cdot y^4}} = \sqrt[4]{\frac{2^4 x^4}{3^4 y^4}}

Next, we can apply the quotient rule to simplify the expression further.

24x434y44=24x4434y44\sqrt[4]{\frac{2^4 x^4}{3^4 y^4}} = \frac{\sqrt[4]{2^4 x^4}}{\sqrt[4]{3^4 y^4}}

Now, we can apply the power rule to simplify the expression even further.

24x4434y44=2x3y\frac{\sqrt[4]{2^4 x^4}}{\sqrt[4]{3^4 y^4}} = \frac{2 x}{3 y}

Conclusion

In this article, we provided a Q&A guide to help you better understand the concept of simplifying radical expressions. We answered some common questions and provided examples to illustrate the concepts.

Final Answer

The final answer is $\boxed{\frac{2 x}{3 y}}$.