How to solve the following problems with exponent?

Question

1. If $9^{x+2}= 240+9^x$ then x= ?
2. $10^x = 64$ what is the value of $10^{(x/2)+1} = ?$
3. $x/x^{1.5} = 8*x^{-1}$ and x > 0 , then x = ?
4. $x^{-2} = 64$, then $x^{1/3} + x^0$ = ?
5. $4^x - 4^{x-1} = 24 $ then $(2x)^x = ?$

Answer 1

$1: 9^x(9^2-1)=240\implies 9^x=3=9^{\frac12}\implies x=\frac12$

$2: 10^x=64\implies 10^{\frac x2+1}=10\cdot (10^x)^{\frac12}=10\cdot (64)^{\frac12}$

$3:\frac x{x^{1.5}}=8\cdot x^{-1}\implies x^{1+1-1.5}=8\implies x^{\frac12}=8\implies x=(8)^2$

$4: x^{-2}=64\implies x^2=(x^{-2})^{-1}=(64)^{-1}=\frac1{64}\implies x=\pm\frac18$

So, $x^{\frac13}+x^0=\left(\frac18\right)^{\frac13}+1=1+\frac1{8^{\frac13}}$

Do you know about the cube roots of $1?$

$5:4^{x-1}(4-1)=24\implies (2^2)^{(x-1)}=8=2^3\implies 2(x-1)=3$

Find $x$ and proceed

The formulas used:

$a^m\cdot a^n=a^{m+n}$

$a^m=a^n\implies m=n$ if $a\ne0,\pm1$

$a^{-m}=\frac1{a^m}$

$a^0=1$ if $a\ne0$