x^{-\frac{2}{3}}=7\frac{1}{9}

Sure, let's solve the equation step by step.

The given equation is: x^(-2/3) = 7 1/9

First, let's convert the mixed number 7 1/9 to an improper fraction.

7 1/9 = 64/9

So, the equation becomes: x^(-2/3) = 64/9

To get rid of the negative exponent, we can take the reciprocal of both sides:

x^(2/3) = 9/64

Now, to solve for x, we need to get rid of the fraction in the exponent. We can do this by raising both sides of the equation to the power of 3/2:

(x^(2/3))^(3/2) = (9/64)^(3/2)

This simplifies to:

x = (9/64)^(3/2)

To simplify further, we can take the square root of the numerator and denominator separately, and then cube the result:

x = ((√9)/ (√64))^3

x = (3/8)^3

x = 27/512

So, the solution to the equation x^(-2/3) = 7 1/9 is x = 27/512.