Let's break down the expression:

$\frac{1}{4} \text{ power of } 2 + \frac{-1}{2} \text{ power of } 3$

This expression can be rewritten mathematically as:

$2^{\frac{1}{4}} + 3^{-\frac{1}{2}}$

Now, let's evaluate each term:

1. $2^{\frac{1}{4}}$ is the fourth root of 2.
2. $3^{-\frac{1}{2}}$ is the negative square root of 3, which can be rewritten as $\frac{1}{\sqrt{3}}$.

Calculations:

1. $2^{\frac{1}{4}} \approx 1.1892$
2. $3^{-\frac{1}{2}} \approx \frac{1}{\sqrt{3}} \approx 0.5774$

So, the expression becomes:

$2^{\frac{1}{4}} + 3^{-\frac{1}{2}} \approx 1.1892 + 0.5774 = 1.7666$

Therefore, the value of the expression is approximately 1.7666.

Would you like more details or have any questions?

Here are some related questions:

1. What is the cube root of 2?
2. How do you calculate fractional exponents?
3. What is the value of $5^{\frac{1}{3}}$?
4. How do negative exponents affect the value of a number?
5. How does $2^{\frac{1}{2}}$ compare to $3^{\frac{1}{2}}$?

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