which expression is equivalent to (x superscript 27 baseline y) superscript one-third?

2025

2 min read

·

31-03-2025

16

0

Share

This article will explore how to simplify the expression (x²⁷y)^(1/3) and find an equivalent expression. We'll break down the process step-by-step, making it easy to understand. Understanding exponent rules is crucial for simplifying algebraic expressions.

Understanding Exponent Rules

Before we tackle the simplification, let's review some key exponent rules:

• Power of a Product Rule: (ab)^n = a^n * b^n. This rule states that when a product is raised to a power, each factor is raised to that power.
• Power of a Power Rule: (a^m)n = a^(m*n). This rule explains that when a power is raised to another power, you multiply the exponents.

Simplifying (x²⁷y)^(1/3)

Now, let's apply these rules to our expression (x²⁷y)^(1/3):

1. Apply the Power of a Product Rule: We can rewrite the expression as (x²⁷)^(1/3) * y^(1/3). This separates the x and y terms, making simplification easier.

2. Apply the Power of a Power Rule: Now focus on (x²⁷)^(1/3). Using the power of a power rule, we multiply the exponents: 27 * (1/3) = 9. Therefore, (x²⁷)^(1/3) simplifies to x⁹.

3. Combine the results: Putting it all together, our simplified expression becomes x⁹y^(1/3). This can also be written as x⁹∛y (where ∛ represents the cube root).

Equivalent Expressions

Therefore, the equivalent expressions to (x²⁷y)^(1/3) are:

• x⁹y^(1/3)
• x⁹∛y

These expressions are mathematically equivalent to the original expression. The choice of which form to use often depends on the context of the problem or personal preference. Both clearly represent the simplified version after applying the relevant exponent rules.

Practice Problems

To solidify your understanding, try simplifying these expressions using the same principles:

1. (a¹²b⁶)^(1/2)
2. (m¹⁵n³)^(1/3)
3. (p⁴⁸q¹²)^(1/4)

Remember to apply the power of a product rule first, then the power of a power rule to each term. This will lead you to the equivalent, simplified form. Checking your work with a calculator or online simplification tool can be helpful.

Conclusion

Simplifying expressions involving exponents requires a solid understanding of exponent rules. By applying these rules systematically—specifically, the power of a product rule and the power of a power rule—we can simplify complex expressions and find equivalent, often more manageable, forms. Understanding this process is fundamental for success in algebra and higher-level mathematics. Remember, (x²⁷y)^(1/3) is equivalent to both x⁹y^(1/3) and x⁹∛y.