How do you simplify sqrt2^109 * sqrtx^306 * sqrtx^11?

${2}^{54} {x}^{158} \sqrt{2 x}$

Explanation:

We have ${\left(\sqrt{2}\right)}^{109} \times {\left(\sqrt{x}\right)}^{306} \times {\left(\sqrt{x}\right)}^{11}$

We can write this as ${2}^{\frac{109}{2}} \times {x}^{\frac{306}{2}} \times {x}^{\frac{11}{2}}$

We can use the rule ${x}^{a} \times {a}^{b} = {x}^{a + b}$ to combine the $x$ terms:

${2}^{\frac{109}{2}} \times {x}^{\frac{306}{2} + \frac{11}{2}}$

${2}^{\frac{109}{2}} \times {x}^{\frac{317}{2}}$

${2}^{54} \times {2}^{\frac{1}{2}} \times {x}^{158} \times {x}^{\frac{1}{2}}$

${2}^{54} {x}^{158} \sqrt{2 x}$

If there is a desire to evaluate ${2}^{54}$, it's roughly $1.8 \times {10}^{16}$