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

Mar 10, 2025 by ADMIN 304 views

**Simplifying Radical Expressions: A Step-by-Step Guide**

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 smaller 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}}

This expression involves a cube root, which is a radical sign with an index of 3. The expression inside the cube root is a fraction, with the numerator being $12 x^2$ and the denominator being $16 y$ .

Step 1: Simplify the Fraction

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 12 and 16 is 4.

\frac{12 x^2}{16 y} = \frac{3 x^2}{4 y}

Step 2: Simplify the Cube Root

Now that we have simplified the fraction, we can simplify the cube root. To do this, we can rewrite the cube root as a product of cube roots.

\sqrt[3]{\frac{3 x^2}{4 y}} = \sqrt[3]{\frac{3}{4}} \cdot \sqrt[3]{x^2} \cdot \sqrt[3]{\frac{1}{y}}

Step 3: Simplify the Cube Roots

Now that we have rewritten the cube root as a product of cube roots, we can simplify each cube root individually.

\sqrt[3]{\frac{3}{4}} = \frac{\sqrt[3]{3}}{\sqrt[3]{4}}

\sqrt[3]{x^2} = x

\sqrt[3]{\frac{1}{y}} = \frac{1}{\sqrt[3]{y}}

Step 4: Combine the Simplified Cube Roots

Now that we have simplified each cube root individually, we can combine them to get the final simplified form.

\frac{\sqrt[3]{3}}{\sqrt[3]{4}} \cdot x \cdot \frac{1}{\sqrt[3]{y}} = \frac{x}{\sqrt[3]{4y^3}} \cdot \sqrt[3]{3}

Step 5: Simplify the Expression

Now that we have combined the simplified cube roots, we can simplify the expression further.

\frac{x}{\sqrt[3]{4y^3}} \cdot \sqrt[3]{3} = \frac{x \sqrt[3]{3}}{\sqrt[3]{4y^3}}

Conclusion

In conclusion, the simplified form of the given expression is:

\frac{x \sqrt[3]{3}}{\sqrt[3]{4y^3}}

This expression involves a cube root, which is a radical sign with an index of 3. The expression inside the cube root is a fraction, with the numerator being $x \sqrt[3]{3}$ and the denominator being $\sqrt[3]{4y^3}$ .

Comparison with Other Options

Now that we have simplified the expression, we can compare it with the other options.

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

This option is not simplified, and it involves a cube root with a fraction inside.

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

This option is not simplified, and it involves a cube root with a fraction inside.

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

This option is not simplified, and it involves a cube root with a fraction inside.

Conclusion

In conclusion, the simplified form of the given expression is:

\frac{x \sqrt[3]{3}}{\sqrt[3]{4y^3}}

Final Answer

The final answer is:

\boxed{\frac{x \sqrt[3]{3}}{\sqrt[3]{4y^3}}}$<br/> **Simplifying Radical Expressions: A Q&A Guide** =====================================================

Introduction

Q&A Section

Q: What is the simplified form of the expression $\sqrt[3]{\frac{12 x^2}{16 y}}$ ?

A: The simplified form of the expression is $\frac{x \sqrt[3]{3}}{\sqrt[3]{4y^3}}$ .

Q: How do I simplify a cube root with a fraction inside?

A: To simplify a cube root with a fraction inside, you can divide both the numerator and the denominator by their greatest common divisor (GCD). Then, you can rewrite the cube root as a product of cube roots and simplify each cube root individually.

Q: What is the greatest common divisor (GCD) of 12 and 16?

A: The greatest common divisor (GCD) of 12 and 16 is 4.

Q: How do I simplify a cube root with a variable inside?

A: To simplify a cube root with a variable inside, you can use the properties of exponents to rewrite the variable as a power of a number. Then, you can simplify the cube root using the properties of radicals.

Q: What is the simplified form of the expression $\sqrt[3]{6 x^2 y^2}$ ?

A: The simplified form of the expression is $x \sqrt[3]{6y^2}$ .

Q: How do I simplify a cube root with a variable and a constant inside?

A: To simplify a cube root with a variable and a constant inside, you can use the properties of exponents to rewrite the variable as a power of a number. Then, you can simplify the cube root using the properties of radicals.

Q: What is the simplified form of the expression $\sqrt[3]{12 x^2 y}$ ?

A: The simplified form of the expression is $\frac{\sqrt[3]{12x^2y}}{2y}$ .

Q: How do I simplify a cube root with a fraction inside and a variable and a constant outside?

A: To simplify a cube root with a fraction inside and a variable and a constant outside, you can divide both the numerator and the denominator by their greatest common divisor (GCD). Then, you can rewrite the cube root as a product of cube roots and simplify each cube root individually.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill to master in mathematics. By understanding the properties of radicals and exponents, you can simplify complex expressions and arrive at the final answer. Remember to always follow the order of operations and to simplify each part of the expression individually.

Final Tips

Always simplify the expression inside the radical sign first.
Use the properties of exponents to rewrite variables as powers of numbers.
Simplify each part of the expression individually.
Use the properties of radicals to simplify the expression.

Common Mistakes

Not simplifying the expression inside the radical sign first.
Not using the properties of exponents to rewrite variables as powers of numbers.
Not simplifying each part of the expression individually.
Not using the properties of radicals to simplify the expression.