Find the sum of all the integer values of m that makes the following equation true: \(\left(2^m3^5\right)^m 9^7=\dfrac{\left(256\cdot3^m\right)^m}{\left(\sqrt2\right)^{14}}\).

tommarvoloriddle Jun 25, 2019

#1**+1 **

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?

Guest Jun 25, 2019

#3**+2 **

(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

CPhill Jun 25, 2019