Simplification and Approximation

Using binomial approximation (1-x)^n is approximately 1-nx for small x. (1-0.01)^(2/3) is approximately 1 - (2/3)*0.01 = 1 - 0.00666 = 0.99333.

3.03^3 = (3 + 0.03)^3 = 3^3 + 3(3^2)(0.03) + 3(3)(0.03^2) + 0.03^3 = 27 + 0.81 + 0.0081 + 0.000027 = 27.818127. Dividing by 27 gives 1.030301, which is approximately 1.03.

Using binomial approximation (x+h)^n = x^n + n*x^(n-1)h. Here (9+0.01)^(1/2) = 9^(1/2) + (1/2)*9^(-1/2)*0.01 = 3 + (1/2)(1/3)*0.01 = 3 + 0.01/6 = 3 + 0.001666... = 3.001666... which rounds to 3.0017.

Using the binomial approximation (1+x)^n approx 1+nx for small x. (-1 + 0.01)^(1/3) = -1 * (1 - 0.01)^(1/3) approx -1 * (1 - 0.01/3) = -1 * (1 - 0.00333) = -0.99666.

1 / sqrt(3) = sqrt(3) / 3. Given sqrt(3) = 1.732, then 1.732 / 3 = 0.577333...

Numerator = 1 + sqrt(0.01) = 1 + 0.1 = 1.1. Denominator = 1 - sqrt(0.1) approx 1 - 0.316 = 0.684. 1.1 / 0.684 approx 1.608.

log10(4.04) = log10(4 * 1.01) = log10(4) + log10(1 + 0.01). Using approximation log10(1+x) approx x * log10(e), we get 0.6021 + 0.01 * 0.4343 = 0.6021 + 0.004343 = 0.606443. The digits are 6064.

6.06^2 is approx 36. 36 - 30.75 = 5.25. The closest integer is 6.

1500.99 / 50.64 is approximately 1500 / 50 = 30.

55% of 2245 = 0.55 * 2245. 0.5 * 2245 = 1122.5. 0.05 * 2245 = 112.25. Sum = 1234.75. The closest option is 1200.

59% of 71080 is approximately 0.6 * 71000 = 42600. 42000 is the closest option.

The square root of 11 is approximately 3.31662479. Rounding this to five decimal places yields 3.31662.