The expression given in the image is:

$\left( x^{-1} \cdot y^{\frac{2}{3}} \cdot z^{-\frac{2}{3}} \right)^{\frac{1}{2}}$

We need to simplify this expression using rational exponents. Let's break it down step by step:

Step 1: Apply the power of a power rule

The power rule states that $(a^m)^n = a^{m \cdot n}$. So, we will apply this rule to each part of the expression:

$\left( x^{-1} \right)^{\frac{1}{2}} = x^{- \frac{1}{2}}$ $\left( y^{\frac{2}{3}} \right)^{\frac{1}{2}} = y^{\frac{2}{3} \cdot \frac{1}{2}} = y^{\frac{1}{3}}$ $\left( z^{-\frac{2}{3}} \right)^{\frac{1}{2}} = z^{-\frac{2}{3} \cdot \frac{1}{2}} = z^{-\frac{1}{3}}$

Step 2: Rewrite the entire expression

Now, we rewrite the expression after applying the exponents:

$x^{-\frac{1}{2}} \cdot y^{\frac{1}{3}} \cdot z^{-\frac{1}{3}}$

Final Answer:

$x^{-\frac{1}{2}} \cdot y^{\frac{1}{3}} \cdot z^{-\frac{1}{3}}$

This is the simplified expression using rational exponents.

Let me know if you'd like further details or have any questions!

Relative Questions:

1. What are the general rules for multiplying exponents with the same base?
2. How do you handle negative exponents in expressions?
3. How do rational exponents relate to radicals?
4. Can you simplify expressions involving fractional exponents further?
5. What is the difference between $\sqrt{x}$ and $x^{\frac{1}{2}}$?

Tip:

When working with negative exponents, remember that $a^{-n} = \frac{1}{a^n}$. You can always rewrite the expression as fractions if needed.