The problem asks us to simplify the expression $(x^{1/7} y^{-1/9}) (x^{7/2} y^{1/2})$ and express the answer with positive exponents.

1. Problem Description

The problem asks us to simplify the expression $(x^{1/7} y^{-1/9}) (x^{7/2} y^{1/2})$ and express the answer with positive exponents.

2. Solution Steps

First, we multiply the terms with the same base by adding their exponents:

$(x^{1/7} y^{-1/9}) (x^{7/2} y^{1/2}) = x^{1/7} x^{7/2} y^{-1/9} y^{1/2}$

$x^{1/7 + 7/2} y^{-1/9 + 1/2}$

We need to find a common denominator for the exponents. For the exponent of x, the common denominator for 7 and 2 is

1

4. For the exponent of y, the common denominator for 9 and 2 is

1

8.

$x^{\frac{1}{7} \cdot \frac{2}{2} + \frac{7}{2} \cdot \frac{7}{7}} y^{\frac{-1}{9} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{9}{9}}$

$x^{\frac{2}{14} + \frac{49}{14}} y^{\frac{-2}{18} + \frac{9}{18}}$

$x^{\frac{2+49}{14}} y^{\frac{-2+9}{18}}$

$x^{\frac{51}{14}} y^{\frac{7}{18}}$

Since both exponents are positive, we don't need to change anything.

3. Final Answer

$x^{51/14} y^{7/18}$