A GP has a first term of a, a common ratio of r and its 6th term is 768. Another GP has a first term of a, a common ratio of 6r and its 3rd term is 3456. Evaluate a and r.

Given: First term is $a$, common ratio is $r$, and the 6th term is 768.
Using the formula $T_n = ar^{n-1}$:

$$T_6 = ar^{6-1}$$ $$768 = ar^5 \quad (*)$$

Given: First term is $a$, common ratio is $6r$, and the 3rd term is 3456.
Using the formula $T_n = ar^{n-1}$:

$$T_3 = a(6r)^{3-1}$$ $$3456 = a(6r)^2$$ $$3456 = a \cdot 36r^2 \quad (**)$$

From equation $(*)$, express $a$ in terms of $r$:

$$a = \frac{768}{r^5}$$

Substitute this expression for $a$ into equation $(**)$:

$$3456 = \left(\frac{768}{r^5}\right) \cdot 36r^2$$ $$3456 = \frac{768 \times 36}{r^3}$$

Multiply both sides by $r^3$:

$$3456r^3 = 768 \times 36$$ $$3456r^3 = 27648$$

Divide by 3456:

$$r^3 = \frac{27648}{3456}$$ $$r^3 = 8$$

Take the cube root of both sides:

$$r = \sqrt[3]{8}$$ $$r = 2$$

$$768 = a(2)^5$$ $$768 = a \cdot 32$$

Divide by 32:

$$a = \frac{768}{32}$$ $$a = 24$$

The values are $a = \boxed{24}$ and $r = \boxed{2}$.