What Values Of $c$ And $d$ Make The Equation True? \sqrt[3]{162 X^c Y^5} = 3 X^2 Y \left(\sqrt[3]{6 Y^d}\right ]A. C = 2 , D = 2 C = 2, D = 2 C = 2 , D = 2 B. C = 2 , D = 4 C = 2, D = 4 C = 2 , D = 4 C. C = 6 , D = 2 C = 6, D = 2 C = 6 , D = 2 D. C = 6 , D = 4 C = 6, D = 4 C = 6 , D = 4

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Introduction

In this article, we will explore the values of $c$ and $d$ that make the given equation true. The equation involves cube roots and exponents, and we will use algebraic manipulation to simplify and solve for the unknown values.

The Given Equation

The given equation is:

$\sqrt[3]{162 x^c y^5} = 3 x^2 y \left(\sqrt[3]{6 y^d}\right)$

Step 1: Simplify the Cube Roots

To simplify the cube roots, we can raise both sides of the equation to the power of 3. This will eliminate the cube roots and allow us to work with exponents.

$\left(\sqrt[3]{162 x^c y^5}\right)^3 = \left(3 x^2 y \left(\sqrt[3]{6 y^d}\right)\right)^3$

Step 2: Expand the Exponents

Using the property of exponents that $(a^m)^n = a^{mn}$, we can expand the exponents on both sides of the equation.

$162 x^c y^5 = 27 x^6 y^3 \cdot 6^{1/3} y^{d/3}$

Step 3: Simplify the Right-Hand Side

We can simplify the right-hand side of the equation by combining the constants and the exponents.

$162 x^c y^5 = 162 x^6 y^{3 + d/3}$

Step 4: Equate the Exponents

Since the bases of the exponents are the same, we can equate the exponents on both sides of the equation.

$c = 6$

Step 5: Equate the Exponents of $y$

We can also equate the exponents of $y$ on both sides of the equation.

$5 = 3 + \frac{d}{3}$

Step 6: Solve for $d$

To solve for $d$, we can multiply both sides of the equation by 3.

$15 = 9 + d$

Step 7: Simplify the Equation

We can simplify the equation by subtracting 9 from both sides.

$6 = d$

Conclusion

In conclusion, the values of $c$ and $d$ that make the equation true are $c = 6$ and $d = 4$.

Final Answer

The final answer is:

• $c = 6$
• $d = 4$

Discussion

The given equation involves cube roots and exponents, and we used algebraic manipulation to simplify and solve for the unknown values. The values of $c$ and $d$ that make the equation true are $c = 6$ and $d = 4$. This problem requires a good understanding of algebraic manipulation and exponent rules.

Related Problems

• Simplifying expressions with exponents
• Equating exponents with different bases
• Solving equations with cube roots

Key Concepts

• Algebraic manipulation
• Exponent rules
• Equating exponents

Practice Problems

• Simplify the expression: $\sqrt[3]{8 x^3 y^2}$
• Equate the exponents: $2^x = 4^y$
• Solve the equation: $\sqrt[3]{27 x^3 y^2} = 3 x y \left(\sqrt[3]{9 y^2}\right)$
  

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Introduction

In our previous article, we explored the values of $c$ and $d$ that make the given equation true. The equation involves cube roots and exponents, and we used algebraic manipulation to simplify and solve for the unknown values. In this article, we will answer some frequently asked questions related to the problem.

Q: What is the main concept behind solving this problem?

A: The main concept behind solving this problem is algebraic manipulation and exponent rules. We used these concepts to simplify the equation and solve for the unknown values.

Q: How do I simplify the cube roots in the equation?

A: To simplify the cube roots, you can raise both sides of the equation to the power of 3. This will eliminate the cube roots and allow you to work with exponents.

Q: What is the property of exponents that we used to expand the exponents?

A: The property of exponents that we used is $(a^m)^n = a^{mn}$. This property allows us to expand the exponents on both sides of the equation.

Q: How do I equate the exponents on both sides of the equation?

A: To equate the exponents, you can compare the exponents of the same base on both sides of the equation. Since the bases are the same, the exponents must be equal.

Q: What is the final answer to the problem?

A: The final answer to the problem is $c = 6$ and $d = 4$.

Q: What are some related problems that I can practice?

A: Some related problems that you can practice include simplifying expressions with exponents, equating exponents with different bases, and solving equations with cube roots.

Q: What are some key concepts that I need to understand to solve this problem?

A: Some key concepts that you need to understand to solve this problem include algebraic manipulation, exponent rules, and equating exponents.

Q: How do I practice solving problems like this?

A: To practice solving problems like this, you can try simplifying expressions with exponents, equating exponents with different bases, and solving equations with cube roots. You can also try solving problems that involve similar concepts and techniques.

Conclusion

In conclusion, solving the equation $\sqrt[3]{162 x^c y^5} = 3 x^2 y \left(\sqrt[3]{6 y^d}\right)$ requires a good understanding of algebraic manipulation and exponent rules. By simplifying the cube roots, expanding the exponents, and equating the exponents, we can solve for the unknown values of $c$ and $d$. We hope that this article has helped you to understand the concepts and techniques involved in solving this problem.

Final Answer

The final answer is:

• $c = 6$
• $d = 4$

Discussion

The given equation involves cube roots and exponents, and we used algebraic manipulation to simplify and solve for the unknown values. The values of $c$ and $d$ that make the equation true are $c = 6$ and $d = 4$. This problem requires a good understanding of algebraic manipulation and exponent rules.

Related Problems

• Simplifying expressions with exponents
• Equating exponents with different bases
• Solving equations with cube roots

Key Concepts

• Algebraic manipulation
• Exponent rules
• Equating exponents

Practice Problems

• Simplify the expression: $\sqrt[3]{8 x^3 y^2}$
• Equate the exponents: $2^x = 4^y$
• Solve the equation: $\sqrt[3]{27 x^3 y^2} = 3 x y \left(\sqrt[3]{9 y^2}\right)$