At a height equal to the radius of the Earth, the gravitational force becomes one-fourth of its value at the surface.

Taking the cube root of both sides, a = 27^(1/3) = 3.

Since 27 = 3^3, we have a = 3.

Substituting a = b^k into a^x = b^y gives (b^k)^x = b^y, leading to kx = y, hence x = ky.

From a^x = b^y, taking log_a on both sides gives log_a(b^y) = x, hence y * log_a(b) = x, thus log_a(b) = y/x.

The power set of a set with n elements has 2^n subsets. Here, n = 4, so the power set has 2^4 = 16 subsets.

The number of subsets of a set with n elements is 2^n. Here, n=2, so the number of subsets is 2^2 = 4.

The power set of B = {a, b} is {∅, {a}, {b}, {a, b}}.

det(B) = (2*4) - (3*1) = 8 - 3 = 5.

C · D = 0*3 + 1*4 + 2*5 = 0 + 4 + 10 = 14.

C · D = 3*0 + (-1)*4 + 2*(-1) = 0 - 4 - 2 = -6.

C · D = 3*0 + (-1)*4 + 2*(-3) = 0 - 4 - 6 = -10.

C · D = 3*4 + (-1)*2 + 2*1 = 12 - 2 + 2 = 12.

C · D = 3*0 + (-2)*4 + 1*(-3) = 0 - 8 - 3 = -11.

The total number of subsets is 2^4 = 16. Proper subsets exclude the set itself, so there are 16 - 1 = 15 proper subsets.

The proper subsets of a set exclude the set itself. For C = {1, 2}, the proper subsets are {∅}, {1}, and {2}, totaling 3 subsets.

The set C = {2, 3, 5, 7}. The power set of C includes all subsets, including the empty set, so it is {{}, {2}, {3}, {5}, {7}, {2, 3}, {2, 5}, {2, 7}, {3, 5}, {3, 7}, {5, 7}, {2, 3, 5}, {2, 3, 7}, {2, 5, 7}, {3, 5, 7}, {2, 3, 5, 7}}.

The even integers less than 10 are 0, 2, 4, 6, and 8.

The power set of an infinite set is also infinite. Since C contains all even integers, its power set is infinite.

The set {2, 4, 6} contains only even integers, making it a subset of C.

The angle A for which cos A = 1/2 is 60°.

Using the identity sin^2 B + cos^2 B = 1, we have sin B = sqrt(1 - (1/2)^2) = sqrt(3/4) = √3/2.

cos(x) = 0 at x = 90° + k*180°, where k is any integer.

tan(x) = sin(x)/cos(x) is undefined when cos(x) = 0.

cos(x) = 0 at x = π/2 + nπ, where n is any integer.

cos(30°) = 0.5.

Using the identity sin^2(x) + cos^2(x) = 1, we have sin^2(x) = 1 - (1/2)^2 = 1 - 1/4 = 3/4. Therefore, sin(x) = √(3/4) = √3/2.

The angles where cos(α) = 1/2 in the interval [0, 2π] are α = π/3 and α = 5π/3.

cos(θ) = 1/2 at θ = 30° and 150°.

cos(θ) = 1/2 at θ = 60° and θ = 300°.