Question 171801
Note: remember, *[Tex \LARGE \sqrt{x^2}=\left(x^{\frac{1}{2}}\right)]



*[Tex \LARGE \sqrt{2^{109}}\cdot\sqrt{x^{306}}\cdot\sqrt{x^{11}}] Start with the given expression



*[Tex \LARGE \sqrt{2\cdot2^{108}}\cdot\sqrt{x^{306}}\cdot\sqrt{x\cdot x^{10}}] Factor {{{2^(109)}}} into {{{2*2^(108)}}} and factor {{{x^(11)=x*x^(10)}}}



*[Tex \LARGE \sqrt{2}\cdot\sqrt{2^{108}}\cdot\sqrt{x^{306}}\cdot\sqrt{x\cdot x^{10}}] Break up *[Tex \LARGE \sqrt{2\cdot2^{108}}] to get *[Tex \LARGE \sqrt{2}\cdot\sqrt{2^{108}}]



*[Tex \LARGE \sqrt{2}\cdot\sqrt{2^{108}}\cdot\sqrt{x^{306}}\cdot\sqrt{x}\cdot\sqrt{x^{10}}] Break up *[Tex \LARGE \sqrt{x\cdot x^{10}}] to get *[Tex \LARGE \sqrt{x}\cdot\sqrt{x^{10}}]



*[Tex \LARGE \sqrt{2}\cdot\left(2^{108}\right)^{\frac{1}{2}}\cdot\left(x^{306}\right)^{\frac{1}{2}}\cdot\sqrt{x}\cdot\left(x^{10}\right)^{\frac{1}{2}}]  Convert from radical notation to exponential notation (see note above).




*[Tex \LARGE \sqrt{2}\cdot\left(2^{\frac{108}{2}}\right)\cdot\left(x^{\frac{306}{2}}\right)\cdot\sqrt{x}\cdot\left(x^{\frac{10}{2}}\right)] Multiply the exponents



*[Tex \LARGE \sqrt{2}\cdot\left(2^{54}\right)\cdot\left(x^{153}\right)\cdot\sqrt{x}\cdot\left(x^{5}\right)] Reduce



*[Tex \LARGE 2^{54}\left(x^{153}\right)\cdot\left(x^{5}\right)\sqrt{2x}] Rearrange the terms.



*[Tex \LARGE 2^{54}x^{158}\sqrt{2x}] Multiply {{{x^(153)}}} and {{{x^(5)}}} to get {{{x^(153)*x^(5)=x^(153+5)=x^(158)}}}




So *[Tex \LARGE \sqrt{2^{109}}\cdot\sqrt{x^{306}}\cdot\sqrt{x^{11}}] simplifies to *[Tex \LARGE 2^{54}x^{158}\sqrt{2x}] 



In other words *[Tex \LARGE \sqrt{2^{109}}\cdot\sqrt{x^{306}}\cdot\sqrt{x^{11}}=2^{54}x^{158}\sqrt{2x}] where {{{x>=0}}}