Final answer:
To solve (1/2)^-2 divided by third root of 2 equals 2^n, we simplify (1/2)^-2 to 4, divide 4 by the cube root of 2 to get 2^(5/3), and conclude that n equals 5/3.
Step-by-step explanation:
To solve the equation (1/2)^-2 \divide \sqrt[3]{2} = 2^n, we can use the properties of exponents to simplify the left side of the equation. Let's start by simplifying (1/2)^-2. To simplify a negative exponent, we can take the reciprocal of the base and change the sign of the exponent, so (1/2)^-2 becomes 2^2, which equals 4.
Next, we have to divide 4 by \sqrt[3]{2}, which is the cube root of 2. The cube root of 2 can also be written as 2^(1/3). Therefore, we have 4 \divide 2^(1/3) or 2^2 \divide 2^(1/3). When dividing expressions with the same base, we subtract the exponents, so 2^2 \divide 2^(1/3) becomes 2^(2 - 1/3) which simplifies to 2^(5/3).
We are given that this expression equals 2^n, so 2^(5/3) = 2^n. Because the bases are the same, the exponents must be equal, so n = 5/3.