Write the two differences using powers of $a$. Both expressions contain a common factor, so their ratio simplifies greatly.

The ratio gives a value for $a^2$. Use the fact that $a$ is positive before substituting into $h(2)$ and then $h(1)$.

Algebra is faster because subtracting the function values reveals a simple ratio. Use Desmos to verify the arithmetic after identifying that the ratio of the two differences equals $a^2$.

Enter $((50-18)/(18-10))$. Desmos displays $4$, confirming that $a^2=4$ and therefore $a=2$ because $a$ is positive.

Enter $10-2$ to find $b$, then enter $2^(1/2)+8$ to evaluate $h(1)$.

Use the given function values to subtract outputs: $h(6)-h(2)=18-10=8$ and $h(10)-h(6)=50-18=32$. From the function rule, these differences are $a^3-a=a(a^2-1)$ and $a^5-a^3=a^3(a^2-1)$, respectively.

Divide the second difference by the first: $\frac{a^3(a^2-1)}{a(a^2-1)}=\frac{32}{8}$. Thus, $a^2=4$. Since $a$ is positive, $a=2$.

Use $h(2)=10$: $2^{2/2}+b=10$, so $2+b=10$ and $b=8$. Therefore, $h(1)=2^{1/2}+8=8+\sqrt{2}$.