To calculate 1/8 to the power of -2/3, or `(1/8)^(-2/3)` , you need to know the following rules of exponents:
The fractional exponent m/n can be rewritten as
`a^(n/m) = root(m) (a^n) = (root(m) a)^n` . In other words, the numerator n of the fraction stays the power of base a, but the denominator m becomes the index of the radical, or mth degree root of a^n.
The negative exponent can be rewritten as
`a^(-n) = 1/a^n` . In other words, negative exponent -n is the positive exponent n of the reciprocal of base a.
These rules can be applied in any order. First, we can rewrite the given expression without the negative exponent. It will be positive power 2/3 of the reciprocal of 1/8, which is 8:
`(1/8)^(-2/3) = 8^(2/3)`
Next, rewrite the fractional exponent as a radical. It will be the third degree (cube) root of 8 squared:
`8^(2/3) = root(3) (8^2) = (root(3) 8)^2`
As you can see, you can either square 8 first, and then take cube root of 8 squared, or you can take cube root of 8 first, and then square the result. Usually, it is easier to take the root first.
By inspection, you can see that cube root of 8 is 2: `root(3) 8 = 2` . This is because the third power of 2 is 8: `2^3 = 2*2*2 = 8` .
Finally, square the cube root of 8: `(root(3) 8)^2 = 2^2 = 4` .
The final answer is 4.