https://file.aiquickdraw.com/mathbot/file/1726804337633kcfhi1rh.jpg

The given expression is:

\[
\frac{\left(16x^2y^{-1/3}\right)^{3/4}}{\left(xy^2\right)^{1/4}}
\]

To simplify this, let's go step by step.

### Step 1: Apply the exponents to each term inside the parentheses.
Starting with the numerator:

\[
\left(16x^2y^{-1/3}\right)^{3/4} = 16^{3/4} \cdot (x^2)^{3/4} \cdot (y^{-1/3})^{3/4}
\]

Now, simplify each part:
- \( 16^{3/4} = (2^4)^{3/4} = 2^3 = 8 \)
- \( (x^2)^{3/4} = x^{2 \cdot \frac{3}{4}} = x^{3/2} \)
- \( (y^{-1/3})^{3/4} = y^{-\frac{1}{3} \cdot \frac{3}{4}} = y^{-1/4} \)

So, the numerator simplifies to:
\[
8x^{3/2}y^{-1/4}
\]

### Step 2: Simplify the denominator.
The denominator is:
\[
\left(xy^2\right)^{1/4} = (x)^{1/4} \cdot (y^2)^{1/4}
\]

Simplifying the exponents:
\[
x^{1/4} \cdot y^{2/4} = x^{1/4} \cdot y^{1/2}
\]

### Step 3: Combine the numerator and the denominator.
Now we have:

\[
\frac{8x^{3/2}y^{-1/4}}{x^{1/4}y^{1/2}}
\]

Simplify the powers of \(x\) and \(y\):
- For \(x\), we subtract the exponents: \( x^{3/2} / x^{1/4} = x^{3/2 - 1/4} = x^{5/4} \)
- For \(y\), we subtract the exponents: \( y^{-1/4} / y^{1/2} = y^{-1/4 - 1/2} = y^{-3/4} \)

So the final simplified expression is:

\[
8x^{5/4}y^{-3/4}
\]

Or equivalently:

\[
\boxed{8x^{5/4}y^{-3/4}}
\]

This is the simplified form of the expression.

Do you have any questions or would you like further details?

---

### Related Questions:
1. How do you simplify expressions with fractional exponents?
2. What is the rule for dividing terms with the same base in exponents?
3. Can fractional exponents be converted to radical expressions?
4. How does raising a product to a fractional power work?
5. What happens when both the base and exponent are negative?

### Tip:
When simplifying exponents, always remember to apply exponent rules consistently: multiplying powers means adding exponents, and dividing powers means subtracting them.

Simplify the expression: (16x^2 * y^(-1/3))^(3/4) / (xy^2)^(1/4)

Learn how to simplify the expression (16x^2 * y^(-1/3))^(3/4) / (xy^2)^(1/4) using fractional exponent rules. This step-by-step solution covers key algebraic concepts, including the application of exponent rules and simplification. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation