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

 Jun 25, 2019
edited by tommarvoloriddle  Jun 25, 2019
edited by tommarvoloriddle  Jun 25, 2019
 #1
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+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?

 Jun 25, 2019
 #2
avatar+1710 
+9

Yes Thank You..... I accidently changed the problem... Whops

tommarvoloriddle  Jun 25, 2019
 #3
avatar+111396 
+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 

 

 

cool cool cool

 Jun 25, 2019
 #4
avatar+1710 
+9

:O Wow Cphill... Long solution... It's very nice!

tommarvoloriddle  Jun 25, 2019

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