Exponents

2^(x-1)+2^(2+x) =144    we can write       

2^x / 2  + 2^2 *2^x  = 144

2^x /2 + 4* 2^x = 144     multiply through by 2

2^x + 8*2^x  = 288

9*2^x  = 288    divide both sides by 9

2^x  = 32       write 32  as 2^5

2^x  = 2^5

So....x  = 5

Solve for x :
2^(x - 1) + 2^(x + 2) = 144

Simplify and substitute y = 2^x.
2^(x - 1) + 2^(x + 2) = (9×2^x)/(2)
 = (9 y)/2:
(9 y)/2 = 144

Multiply both sides by 2/9:
y = 32

Substitute back for y = 2^x:
2^x = 32

32 = 2^5:
2^x = 2^5

Equate exponents of 2 on both sides: 
x = 5

2^(x-1)+2^(2+x)=144

\(\begin{array}{|rcll|} \hline 2^{x-1}+2^{2+x} &=& 144 \\ 2^{x-1}+2^{x+2} &=& 144 \quad & | \quad \cdot 2^3 \\ 2^{x-1}2^3+2^{x+2}2^3 &=& 144 *2^3 \\ 2^{x-1+3}+2^{x+2}*8 &=& 144 * 8 \\ 2^{x+2}+2^{x+2}*8 &=& 144 * 8 \\ 2^{x+2}*9 &=& 144 * 8 \quad & | \quad :9 \\ 2^{x+2} &=& \frac{144 * 8}{9} \\ 2^{x+2} &=& 16*8 \\ 2^{x+2} &=& 2^42^3 \\ 2^{x+2} &=& 2^{4+3} \\ 2^{x+2} &=& 2^{7} \\\\ x+2 &=& 7 \\ x&=& 7-2 \\ \mathbf{x} & \mathbf{=} & \mathbf{5} \\ \hline \end{array}\)