Simplifying Expressions Which Expression Is Equivalent To (x^27 Y)^(1/3)

Jul 11, 2025 by ADMIN 73 views

Understanding Equivalent Expressions for (x^27 * y)^(1/3)

This article delves into the process of simplifying the expression (x^27 * y)^(1/3) and identifying the equivalent expression from a given set of options. We will explore the fundamental rules of exponents and radicals, providing a step-by-step solution to the problem. This guide is designed to help students and math enthusiasts strengthen their understanding of algebraic manipulations and expression simplification.

Breaking Down the Problem: Exponents and Radicals

When dealing with expressions involving exponents and radicals, it is crucial to understand the underlying principles that govern their behavior. In this case, we are presented with the expression (x^27 * y)^(1/3). This expression involves both an exponent (1/3) and variables (x and y). To simplify this expression, we need to apply the power of a product rule and the relationship between fractional exponents and radicals.

The power of a product rule states that (ab)^n = a^n * b^n, where a and b are any real numbers or variables, and n is any exponent. This rule allows us to distribute the exponent outside the parentheses to each factor inside the parentheses. Applying this rule to our expression, we get:

(x^27 * y)^(1/3) = (x²⁷⁾(1/3) * y^(1/3)

Now, we need to simplify each term separately. For the first term, (x²⁷⁾(1/3), we can use the power of a power rule, which states that (a^m)n = a^(m*n), where a is any real number or variable, and m and n are any exponents. Applying this rule, we multiply the exponents:

(x²⁷⁾(1/3) = x^(27 * (1/3)) = x^9

The second term, y^(1/3), involves a fractional exponent. A fractional exponent represents a radical. Specifically, a^(1/n) is equivalent to the nth root of a, denoted as √n. In our case, y^(1/3) is equivalent to the cube root of y, denoted as ³√y. Thus:

y^(1/3) = ³√y

Now, we can combine the simplified terms to get the equivalent expression:

(x^27 * y)^(1/3) = x^9 * ³√y

Therefore, the expression equivalent to (x^27 * y)^(1/3) is x^9(³√y).

Step-by-Step Solution

Let's break down the solution into clear, manageable steps:

Apply the power of a product rule: Distribute the exponent (1/3) to both x^27 and y.

(x^27 * y)^(1/3) = (x²⁷⁾(1/3) * y^(1/3)
Apply the power of a power rule: Multiply the exponents for the term involving x.

(x²⁷⁾(1/3) = x^(27 * (1/3)) = x^9
Convert the fractional exponent to a radical: Rewrite y^(1/3) as the cube root of y.

y^(1/3) = ³√y
Combine the simplified terms: Multiply the simplified terms to obtain the final expression.

x^9 * ³√y = x^9(³√y)

This step-by-step approach ensures a clear understanding of the process involved in simplifying the expression.

Analyzing the Options

Now, let's analyze the given options in the context of our solution.

x^3(³√y): This option is incorrect because the exponent of x should be 9, not 3.
x^9(³√y): This is the correct option, as it matches our simplified expression.
x^27(³√y): This option is incorrect because the exponent of x should be 9, not 27.
x^24(³√y): This option is incorrect because the exponent of x should be 9, not 24.

By comparing each option with our simplified expression, we can confidently identify the correct answer.

Key Concepts and Rules

To master simplifying expressions with exponents and radicals, it's essential to understand the following key concepts and rules:

Power of a product rule: (ab)^n = a^n * b^n
Power of a power rule: (a^m)n = a^(m*n)
Fractional exponents and radicals: a^(1/n) = √n

Understanding these rules will enable you to simplify a wide range of algebraic expressions.

Common Mistakes to Avoid

When simplifying expressions with exponents and radicals, it's important to be aware of common mistakes that students often make. Here are a few to keep in mind:

Incorrectly applying the power of a product rule: Ensure you distribute the exponent to all factors within the parentheses.
Misunderstanding the power of a power rule: Remember to multiply the exponents, not add them.
Confusing fractional exponents and radicals: Be clear on the relationship between a^(1/n) and √n.
Forgetting to simplify completely: Always ensure that the expression is in its simplest form.

By avoiding these common mistakes, you can improve your accuracy and confidence in simplifying algebraic expressions.

Practice Problems

To further solidify your understanding, try simplifying the following expressions:

(a^16 * b)^(1/4)
(p^25 * q¹⁰⁾(1/5)
(m^8 * n¹²⁾(1/2)

Working through these practice problems will help you apply the concepts and rules we've discussed and develop your problem-solving skills.

Real-World Applications

The concepts of exponents and radicals are not just theoretical; they have numerous real-world applications in various fields, including:

Physics: Calculating the velocity and acceleration of objects.
Engineering: Designing structures and systems.
Computer science: Analyzing algorithms and data structures.
Finance: Calculating compound interest and investment growth.

Understanding these mathematical concepts can be invaluable in various professional and practical contexts.

Conclusion

In this article, we've thoroughly explored the process of simplifying the expression (x^27 * y)^(1/3) and identifying the equivalent expression. By understanding the power of a product rule, the power of a power rule, and the relationship between fractional exponents and radicals, we can confidently simplify complex algebraic expressions. Remember to practice regularly and be mindful of common mistakes to achieve mastery in this area. The correct equivalent expression is x^9(³√y). This comprehensive guide should provide you with a solid foundation for tackling similar problems and enhancing your mathematical skills.