A different was of thinking about it is with the log rules. Let's take 2.4% to the hundredth power, which is assuredly more than 240%.

    y = 1.024^100
    ln y = ln (1.024^100) = 100 * ln 1.024

Now, we can approximate ln x = x-1 for values of x near one, so we get

    ln y = 100 * 0.024 = 2.4
    y = exp 2.4 = e ^ 2.4 ~ 2.7^2.4
    2.7^2.4 = (27/10)^2.4 = (3^3/10) ^ 2.4
    2.7^2.4 = (3^3)^2.4 / (10^2.4)
    = 3^7.2 / (100 * 10^0.4)
    = 3*(3^6)*3^0.2 / (100 * 10^0.4)
    = 3*(729/100)*3^0.2/10^0.4
    = 3*(7.29)*(3/100)^0.2
    = 21.87*(0.03)^(1/5)

Finally, (1/2)^5 = 1/32 ~ 0.03

    y ~ 21.87 * 1/2 = 10.94

Pulling out Python, the actual answer is 10.72, so we're within a few percent.