+1712 Find the sum of all the integer values of m that makes the following equation true: \(\left(2^m3^5\right)^m 9^7=\dfrac{\left(256\cdot3^m\right)^m}{\left(\sqrt2\right)^{14}}\).
Let the speed of the Plane =P
Let the speed of the wind = W
P + W = 840 / 1.75...............(1)
P - W = 840 / 2.....................(2)
Can you solve the 2 simultaneous equations?
+128406 (2^m*3^5)^m *9^7 = (256 * 3^m)^m / (2^1/2)^14
(2^m*3^5)^m *(3^2)^7 = (2^8 * 3^m)^m / 2^7
2^(m^2) * 3^(5m) * 3^14 = 2*(8m) * 3^(m^2) / 2^7
2^(m^2) * 3^(5m + 14) = 2^(8m - 7) * 3^(m^2)
Equating exponents on the bases on both sides, we have that
m^2 = 8m - 7 and m^2 = 5m + 14
m^2 - 8m + 7 = 0 m^2 - 5m - 14 = 0
(m -7)(m - 1) = 0 (m + 2)(m - 7) = 0
Set each factor to 0 and solve for m and we have that the possible values forn are
m = 7 m = 1 and m = -2 m = 7
The common solution is that m = 7
