Understanding the Expression
The expression we are dealing with is (81x^4/y^-8)^(1/4). To simplify this expression, we’ll follow the steps of exponent rules and properties.
Step 1: Rewrite Negative Exponents
- The term y^-8 can be rewritten as 1/y^8.
- Thus, the expression becomes: 81x^4 * y^8.
Step 2: Simplify Inside the Parentheses
- Now, we have: (81x^4 * y^8).
Step 3: Apply the Fourth Root
- The entire expression is raised to the power of 1/4. We can distribute this exponent to each factor inside the parentheses:
- (81)^(1/4)
- (x^4)^(1/4)
- (y^8)^(1/4)
Step 4: Calculate Each Component
- Calculating 81^(1/4):
- 81 is 3^4, so (81)^(1/4) = 3.
- Calculating (x^4)^(1/4):
- When multiplying exponents, we multiply their powers: x^(4*(1/4)) = x^1 = x.
- Calculating (y^8)^(1/4):
- Similarly, y^(8*(1/4)) = y^2.
Final Expression
- Combining all components, we get:
3xy^2.
Thus, the simplified form of (81x^4/y^-8)^(1/4) is 3xy^2.