Question

(72*78*85*90*90)/(68*84*91*108) (There's an easier way than multiplying out the giant products 72 78 85 90 98 and 68 84 91 108)HOW?!

Answer 1

Cancel common factors. For instance,

$\dfrac{72}{68}=\dfrac{2^3\cdot3^2}{2^2\cdot17}=\dfrac{2\cdot3^2}{17}=\dfrac{18}{17}$

For this expression, we have

$\dfrac{72\cdot78\cdot85\cdot90^2}{68\cdot84\cdot91\cdot108}=\dfrac{2^6\cdot3^7\cdot5^3\cdot13\cdot17}{2^6\cdot3^4\cdot7^2\cdot13\cdot17}=\dfrac{3^3\cdot5^3}{7^2}=\dfrac{15^3}{7^2}=\dfrac{3375}{49}$