Question

Please help me and show your work!!

Answer 1

$\bf \cfrac{\sqrt{250x^{16}}}{\sqrt{2x}}\qquad \begin{cases} 250=2\cdot 5\cdot 5\cdot 5\\ \qquad 2\cdot 5\cdot 5^2\\ x^{16}=x^{8\cdot 2}\\ \qquad (x^8)^2 \end{cases}\implies \cfrac{\sqrt{2\cdot 5\cdot 5^2\cdot (x^8)^2}}{\sqrt{2\cdot x}}$

$\bf \cfrac{5x^8\sqrt{2\cdot 5}}{\sqrt{2}\cdot \sqrt{x}}\implies \cfrac{5x^8\underline{\sqrt{2}}\cdot \sqrt{5}}{\underline{\sqrt{2}}\cdot \sqrt{x}}\implies \cfrac{5x^8\sqrt{5}}{\sqrt{x}}\impliedby \begin{array}{llll} \textit{and rationalizing the}\\ \textit{denominator} \end{array}$

$\bf \cfrac{5x^8\sqrt{5}}{\sqrt{x}}\cdot \cfrac{\sqrt{x}}{\sqrt{x}}\implies \cfrac{5x^8\sqrt{5}\cdot \sqrt{x}}{(\sqrt{x})^2}\implies \cfrac{5x^8\sqrt{5x}}{x}\implies 5x^8x^{-1}\sqrt{5x} \\\\\\ 5x^{8-1}\sqrt{5x}\implies 5x^7\sqrt{5x}$