Simplify the given expression to rational exponent form and justify each step by identifying the properties of rational exponents used. All work must be shown.

1/cubed root of x to -6 exponent

1/∛x = x^(-1/3)

(x^(-1/3))^-6 = x^((-1/3)(-6)) = x^2

To simplify the given expression, we need to rewrite it in rational exponent form by using the properties of rational exponents.

Step 1: Start with the expression 1/(∛x)^-6.

Step 2: Using the property of rational exponents that says (a/b)^c = a^c / b^c, we can rewrite the expression as 1/ (∛x)^-6 = 1/ [1/(∛x)^6].

Step 3: Next, we can simplify the expression within the brackets by applying the property of rational exponents that states (∛a)^b = a^(b/3). So, (∛x)^6 becomes x^(6/3) = x^2.

Step 4: Now, we can substitute x^2 back into 1/[1/(∛x)^6] to get 1/[1/x^2].

Step 5: Applying the property of rational exponents (1/a)^b = a^-b, we can rewrite the expression as [x^2]^(-1) = x^-2.

Therefore, the simplified expression in rational exponent form is x^-2.

In summary, we simplified the given expression 1/(∛x)^-6 to x^-2 by utilizing the properties of rational exponents.