To simplify the expression [81x^4/y^-8]^1/4, we will apply the rules of exponents step by step.



Inside the brackets, we have 81x^4/y^-8. To simplify this expression, we can apply the following rules:

Rule 1: (a/b)^n = a^n / b^n
Rule 2: (a^n)^m = a^(n * m)

Using Rule 1, we can rewrite the expression as (81x^4)^1 / (y^-8)^1.

Now, using Rule 2, we can simplify further:

(81x^4)^1 = 81^1 * (x^4)^1 = 81 * x^4
(y^-8)^1 = y^(-8 * 1) = y^-8

So, the expression inside the brackets simplifies to 81x^4 / y^-8.



Now, we have the expression 81x^4 / y^-8 raised to the power of 1/4.

To simplify this expression, we can apply the following rule:

Rule 3: (a/b)^n = a^n / b^n

Using Rule 3, we can rewrite the expression as (81x^4)^1/4 / (y^-8)^1/4.

Now, we can apply the power of a power rule:

(a^m)^n = a^(m * n)

Using this rule, we can simplify further:

(81x^4)^1/4 = 81^(1/4) * (x^4)^(1/4) = 3 * x

(y^-8)^1/4 = y^(-8 * 1/4) = y^-2

So, the expression simplifies to 3x / y^-2.



To simplify the expression 3x / y^-2, we can apply the rule:

a / b^-n = a * b^n

Using this rule, we can rewrite the expression as 3x * y^2.

Therefore, the simplified value of [81x^4/y^-8]^1/4 is 3x * y^2.