At height R, g' = g / (2^2) = g / 4.

The standard value of gravitational acceleration at the surface of the Earth is approximately 9.81 m/s^2.

The standard value of gravitational acceleration on the surface of the Earth is approximately 9.8 m/s².

i^4 = (i^2)^2 = (-1)^2 = 1.

For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0, thus k^2 = 64, giving k = -8 or 8. The answer is -4.

The discriminant must be negative: k^2 - 4*1*9 < 0 => k^2 < 36 => |k| < 6, hence k > 6.

If the roots are reciprocals, then k = sum of roots = 0 and product = 1. Thus, k = 2.

For equal roots, the discriminant must be zero: (-4)^2 - 4*1*k = 0 => 16 - 4k = 0 => k = 4.

Setting 2k + 2 = 0 gives k = 2.

Setting k(0) = 0^2 + 1 gives k = 1.

Setting k(2) = 2^2 gives 2k = 4, thus k = 2.

The minimum value occurs at x = -b/(2a) = -2/(2k). Setting f(-1) = -3 gives k = -2.

For no real roots, the discriminant must be less than zero: k^2 - 4*1*16 < 0 => k^2 < 64 => |k| < 8.

For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0, thus k^2 = 64, giving k = -8 or k = 8. The answer is -8.

The discriminant must be less than zero: k^2 - 4*1*16 < 0 => k^2 < 64 => k < 8 and k > -8.

For one real root, the discriminant must be zero: k^2 - 4*1*25 = 0, thus k^2 = 100, giving k = -10 or k = 10.

For no real roots, the discriminant must be less than zero: k^2 - 4*1*9 < 0, thus k > 6.

For one real root, the discriminant must be zero: k^2 - 4*1*9 = 0 => k^2 = 36 => k = ±6.

For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0 => k^2 = 64 => k = 8 or k = -8.

Using Vieta's formulas, k = 3 + 4 = 7.

For d orbitals, the azimuthal quantum number l is equal to 2.

log2(8) = 3 because 2^3 = 8.

log_10(1000) = 3 and log_10(0.01) = -2, thus 3 - 2 = 1.

log_10(1000) = log_10(10^3) = 3.

log_2(32) = 5 and log_2(4) = 2. Therefore, 5 - 2 = 3.

log_2(32) = 5 and log_2(8) = 3. Therefore, 5 - 3 = 2.

Since 32 = 2^5, log_2(32) = 5.

log_3(27) = 3 and log_3(9) = 2. Therefore, 3 - 2 = 1.

Since 27 = 3^3, log_3(27) = 3.

log_3(81) = log_3(3^4) = 4.