Compute \(\frac{\lfloor \sqrt[4]{1} \rfloor \cdot \lfloor \sqrt[4]{3} \rfloor \cdot \lfloor \sqrt[4]{5} \rfloor \dotsm \lfloor \sqrt[4]{2015} \rfloor}{\lfloor \sqrt[4]{2} \rfloor \cdot \lfloor \sqrt[4]{4} \rfloor \cdot \lfloor \sqrt[4]{6} \rfloor \dotsm \lfloor \sqrt[4]{2016} \rfloor}.\)
+128460 Note that
2^4 = 16
3^4 = 81
4^4 = 256
5^4 = 625
6^4 = 1296
7^4 = 2401
So each floor from (1)^.25 to ( 15)^.25 = 1
And floor (16)^.25 = 2
And each floor from (17)^.25 to (80)^.25 = 2
And floor (81)^.25 = 3
And each floor from (82)^.25 to (255)^.25 = 3
And floor (256)^.25 = 4
And each floor from (257)^.25 to (624)^.25 = 4
And floor (625)^.25 = 5
And each floor from (626)^.25 to (1295)^.25 = 5
And floor (1296)^.25 = 6
And each floor from (1297)^.25 to (2016)^.25 = 6
So...note the pattern.....
(1)^8 * (2)^32 * (3)^88 * (4)^184 * .... 1 * 3 * ......
______________________________ = ___________
(1)^7 * (2)^33 * (3)^87* (4)^185 * ....... 2 * 4 * .....
1 * 3 * 5 *
__________ =
2 * 4 * 6
15
____ =
48
5
__
16
+25 I think your calculation is incorrect, the answer I got is \(\frac{1}{\lfloor \sqrt[4]{2016} \rfloor}\)
