How to implement exponentiation of a rational number without nth root?

Question

Its available for me only log(base "e"), sin, tan and sqrt (only square root) functions and the basic arithmetical operators (+ - * / mod). I have also the "e" constant.

I'm experimenting several problems with Deluge (zoho.com) for these restrictions. I must implement exponentiation of rational (fraction) bases and exponents.

Answer 1

Say you want to calculate pow(A, B)

Consider the representation of B in base 2:

B = b[n]   * pow(2, n    ) +
    b[n-1] * pow(2, n - 1) +
    ...
    b[2]   * pow(2, 2    ) +
    b[1]   * pow(2, 1    ) +
    b[0]   * pow(2, 0    ) +
    b[-1]  * pow(2, -1   ) +
    b[-2]  * pow(2, -2   ) +
    ...

 = sum(b[i] * pow(2, i))

where b[x] can be 0 or 1 and pow(2, y) is an integer power of two (i.e., 1, 2, 4, 1/2, 1/4, 1/8).

Then,

pow(A, B) = pow(A, sum(b[i] * pow(2, i)) = mul(pow(A, b[i] * pow(2, i)))

And so pow(A, B) can be calculated using only multiplications and square root operations

Answer 2

If you have a function F() that does e^x, where e is the constant, and x is any number, then you can do this: (a is base, b is exponent, ln is log-e)

a^b = F(b * ln(a))

If you don't have F() that does e^x, then it gets trickier. If your exponent (b) is rational, then you should be able to find integers m and n so that b = m/n, using a loop of some sort. Once you have m and n, you make another loop which multiples a by itself m times to get a^m, then multiples a by itself n times to get a^n, then divide a^m/a^n to get a^(m/n), which is a^b.