Solving $2^x+2^{3x}=16$

Question

Solve $$2^x+2^{3x}=16$$

My attempt:

The simplest form of this equation I could obtained after simplification is $1+2^{2x}=2^{4-x}$. I couldn't find the value of $x$

My conclusion:

Either the question is wrongly framed or there is no solution.

Answer 1

$$2^x+2^{3x}=16$$ Let $u=2^x$ $$\implies u+u^3=16$$ solve $u^3+u-16=0$ using Newton–Raphson you get : $$u\approx 2.38768$$ $$2^x=2.38768$$ $$\boxed{x=1.25561}$$

Answer 2

So this is what I did: hand worked problem

After finding y this should be easy. I followed the cubic formula found in the next link: Cubic Formula

I hope this helps you out.

Answer 3

Plotting both functions, we see that there is a solution on $[1,2]$. My initial feeling is that we can only find the solution by using a computer.