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avatar+94 

Solve for the rational numbers x and y:

\(2^{x+y} \cdot 3^{x-y} \cdot 6^{2x+2y}= 72\)

Show answer in ordered pair. (x, y)

 Nov 3, 2019
 #1
avatar+19937 
+2

(1/2, 1/2)    ?

 Nov 3, 2019
 #2
avatar+94 
+1

yea, you are correct

VooFIX  Nov 3, 2019
 #3
avatar+2482 
+2

Prime factorize

 

72

 

8 * 9 

 

23 * 32

 

Factor out a 62

 

2 * 62

 

Make system of equations

2x + 2y = 2

 

x + y = 1

 

Solving, we get x = 1/2 and y = 1/2

CalculatorUser  Nov 3, 2019
 #4
avatar+105238 
+2

2^(x + y)  * 3^(x - y)  * 6^(2x + 2y)   =  72

 

2^(x + y) *  3^(x - y) * ( 6^2)^(x + y)  =  72

 

2^(x + y) * 3(x - y) * 36(^(x + y)  =  72

 

( 2 * 36)^(x + y)  * 3^(x - y)  =  72

 

(72)^(x + y)  * 3^(x - y)  =  72

 

Since

72^(1) * 3^(0)  =   72

 

This will be true when

 

x +  y   =   1

x -  y  =     0              add these

 

2x   =  1

x   =1/2

 

And  

1/2 + y   = 1

y  = 1/2

 

(x,y)   =  (1/2, 1/2)

 

 

cool cool cool

 Nov 3, 2019

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