https://r2.erweima.ai/mathbot/image/1726708059369iev5tabf.jpg

The expression shown at the top of the image is:

\[
\left( 4x - \sqrt[3]{y} \right)^{\frac{3}{2}}
\]

We are asked to identify the equivalent expression among the four options provided. Let's analyze the expression step by step.

### Breaking Down the Expression:

The given expression is:
\[
\left( 4x - \sqrt[3]{y} \right)^{\frac{3}{2}}
\]

- The outer exponent is \( \frac{3}{2} \), which can be interpreted as: 
  - First, raising the expression inside the parentheses to the power of \( 3 \) (the numerator of the exponent).
  - Then taking the square root of the result (because the denominator of the exponent is 2).

In simpler terms:
\[
\left( 4x - \sqrt[3]{y} \right)^{\frac{3}{2}} = \sqrt{\left( 4x - \sqrt[3]{y} \right)^3}
\]

### Identifying the Correct Option:

The second option (labeled as "Opción 1") matches this form because it shows:
\[
\sqrt{(4x - \sqrt[3]{y})^3}
\]
This is exactly equivalent to the given expression.

### Conclusion:

The correct answer is **Opción 1**.

---

Would you like more details or have any further questions?

Here are 5 related questions to extend this topic:
1. How do you simplify expressions with fractional exponents in general?
2. What is the difference between roots and powers in an expression like this?
3. How would the expression change if the exponent were \( \frac{2}{3} \) instead of \( \frac{3}{2} \)?
4. Can you apply the same steps if the expression inside the parentheses were more complex?
5. How does the cube root of \( y \) affect the final simplification?

**Tip**: When dealing with fractional exponents, remember that the denominator represents the root and the numerator represents the power.

The expression (4x - y^(1/3))^(3/2) is given, and you are asked to select the equivalent form from the options.

Learn how to simplify the expression (4x - y^(1/3))^(3/2) by converting the fractional exponent into a root and power. This problem demonstrates key algebraic skills involving fractional exponents and is suitable for high school students (Grades 9-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation