Maxima can be used as a powerful calucator:

144*17 - 9;

2439

Maxima can compute with very large numbers. The following expample
computes the 25^{th} power of 144:

144^25;

910043815000214977332758527534256632492715260325658624

This is more than a pocket calculator can do!

Now we compute the 25^{th} root of that result:

%^(1/25);

The percent sign is a special variable, its value is always the last result. The arrow is the exponentiation operator. In our input we have to use some elementary mathematical knowledge: We write the root as a power with a fractional exponent. We obtain this answer:

144

But computer algebra is more than just computation with numbers. It is computation with symbols.

Let us play with a polynomial in two variables:

(%i12) (x + 2*y)^4; 4 (%o12) (2 y + x) (%i13) expand(%); 4 3 2 2 3 4 (%o13) 16 y + 32 x y + 24 x y + 8 x y + x (%i14) factor(%); 4 (%o14) (2 y + x)

Maxima can compute derivatives:

(%i15) diff(sin(x)*cos(x), x); 2 2 (%o15) cos (x) - sin (x) (%i16) trigsimp(%); 2 (%o16) 2 cos (x) - 1 (%i17) diff(%, x); (%o17) - 4 cos(x) sin(x) (%i18) diff( sin(x)*cos(x), x, 2); (%o18) - 4 cos(x) sin(x)

Maxima can rewrite trigonometric expressions in a canonical form, namely as finite Fourier sums:

```
(%i19) trigreduce (sin(x)^5);
sin(5 x) - 5 sin(3 x) + 10 sin(x)
(%o19) ---------------------------------
16
```

Maxima can compute indefinite integrals:

integrate((x + 1)/(x^3 - 8), x);

2 x + 2 2 atan(---------) log(x + 2 x + 4) 2 sqrt(3) log(x - 2) - ----------------- + --------------- + ---------- 8 4 sqrt(3) 4

Here is a longer example that shows that Maxima can compute quite complicated integrals and can also often reconstruct the given integrand:

assume(m>4); [m > 4] integrate(x^m*(a + b*x)^3, x); 3 m + 4 2 m + 3 2 m + 2 3 m + 1 b x 3 a b x 3 a b x a x --------- + ------------- + ------------- + --------- m + 4 m + 3 m + 2 m + 1 diff(%, x); 3 m + 3 2 m + 2 2 m + 1 3 m b x + 3 a b x + 3 a b x + a x factor(%); m 3 x (b x + a)