Short Answer 
The answer outlines the rules of exponents needed for simplification, including handling negative exponents, roots, and powers. It details the steps to simplify the expressions 343^{-2/3} to dfrac{1}{49} and dfrac{1}{81^{1/4}} to dfrac{1}{3}, demonstrating the application of these rules.
Step 1: Understand Exponent Rules
Familiarize yourself with the key rules of exponents that will be essential for simplification. The important rules include:
- a^{-b} = dfrac{1}{a^b} – This indicates that a negative exponent inverts the base.
- sqrt[n]{a} = a^{(1/n)} – This shows that taking the nth root is equivalent to raising the number to the power of 1/n.
- (a^b)^c = a^{bc} – This states that when raising a power to another power, multiply the exponents.
Step 2: Simplify Expression 19
Using the exponent rules, we can simplify the expression 343^{-2/3}. Follow these steps:
- Apply the negative exponent rule: 343^{-2/3} = dfrac{1}{(sqrt[3]{343})^2}.
- Calculate the cube root: sqrt[3]{343} = 7.
- Substitute and simplify: dfrac{1}{(7)^2} = dfrac{1}{49}.
Step 3: Simplify Expression 20
Next, simplify the expression dfrac{1}{81^{1/4}} using the exponent rules:
- Rewrite using the fourth root: dfrac{1}{(3^4)^{1/4}}.
- Apply the power of a power rule: (3^4)^{1/4} = 3^{(4 cdot 1/4)} = 3^{1}.
- Finally, substitute and simplify: dfrac{1}{3}.