Exponential Equation (unclear solution): 2^{x+1} + 2^{x-2} = 9/2

[SED]

Question 10's solution:

. . .\(\displaystyle 2^{x+1}\, +\, 2^{x-2}\, =\, \dfrac{9}{2}\)

. . .\(\displaystyle 2^x\, \cdot\, 2\, +\, \dfrac{2^x}{2^2}\, =\, \dfrac{9}{2}\)

. . .\(\displaystyle \dfrac{\left(4\, \cdot\, 2^x\, \cdot\, 2\right)\, +\, 2^x}{4}\, =\, \dfrac{9}{2}\)

. . .\(\displaystyle \dfrac{\left(8\, \cdot\, 2^x\right)\, +\, 2^x}{4}\, =\, \dfrac{9}{2}\)

. . .\(\displaystyle \left(8\, \cdot\, 2^x\right)\, +\, 2^x\, =\, 18\)

Solution from there...

. . .\(\displaystyle \left(8\, \cdot\, 2^x\right)\, +\, 2^x\, =\, 18\)

. . .\(\displaystyle 9\, \cdot\, 2^x\, =\, 18\)

. . .\(\displaystyle \dfrac{9\, \cdot\, 2^x}{9}\, =\, \dfrac{18}{9}\)

. . .\(\displaystyle 2^x\, =\, 2\)

. . .\(\displaystyle x\, =\, 1\)

Question: How to go from here:

. . .\(\displaystyle \left(8\, \cdot\, 2^x\right)\, +\, 2^x\, =\, 18\)

...to here?

. . .\(\displaystyle 9\, \cdot\, 2^x\, =\, 18\)

 

[mmm4444bot]

Question 10's solution:

. . .\(\displaystyle 2^{x+1}\, +\, 2^{x-2}\, =\, \dfrac{9}{2}\)

. . .\(\displaystyle 2^x\, \cdot\, 2\, +\, \dfrac{2^x}{2^2}\, =\, \dfrac{9}{2}\)

. . .\(\displaystyle \dfrac{\left(4\, \cdot\, 2^x\, \cdot\, 2\right)\, +\, 2^x}{4}\, =\, \dfrac{9}{2}\)

. . .\(\displaystyle \dfrac{\left(8\, \cdot\, 2^x\right)\, +\, 2^x}{4}\, =\, \dfrac{9}{2}\)

. . .\(\displaystyle \left(8\, \cdot\, 2^x\right)\, +\, 2^x\, =\, 18\)

Solution from there...

. . .\(\displaystyle \left(8\, \cdot\, 2^x\right)\, +\, 2^x\, =\, 18\)

. . .\(\displaystyle 9\, \cdot\, 2^x\, =\, 18\)

. . .\(\displaystyle \dfrac{9\, \cdot\, 2^x}{9}\, =\, \dfrac{18}{9}\)

. . .\(\displaystyle 2^x\, =\, 2\)

. . .\(\displaystyle x\, =\, 1\)

Question: How to go from here:

. . .\(\displaystyle \left(8\, \cdot\, 2^x\right)\, +\, 2^x\, =\, 18\)

...to here?

. . .\(\displaystyle 9\, \cdot\, 2^x\, =\, 18\)

Your question is how to get the expression:

9 * 2^x

from the expression:

8 * 2^x + 2^x

Think symbolically, and it becomes a combination of like-terms.

That is, if you have eight objects (each one looks like 2^x), and you add one more, then you have a total of nine objects.

In other words:

8C + C = 9C

where symbol C represents 2^x :cool: