9.5 Divide square roots

$\frac{\sqrt{27}}{6}$

$\frac{\sqrt{50}}{10}$

$\frac{\sqrt{72}}{9}$

$\frac{\sqrt{243}}{6}$

$\frac{2-\sqrt{32}}{8}$

$\frac{3+\sqrt{27}}{9}$

$\frac{6+\sqrt{45}}{6}$

$\frac{10-\sqrt{200}}{20}$

$\frac{\sqrt{80}}{\sqrt{125}}$

$\frac{\sqrt{72}}{\sqrt{200}}$

$\frac{\sqrt{128}}{\sqrt{72}}$

$\frac{\sqrt{48}}{\sqrt{75}}$

ⓐ $\frac{\sqrt{8x^6}}{\sqrt{2x^2}}$ ⓑ $\frac{\sqrt{200m^5}}{\sqrt{98m}}$

ⓐ $\frac{\sqrt{10y^3}}{\sqrt{5y}}$ ⓑ $\frac{\sqrt{108n^7}}{\sqrt{243n^3}}$

$\frac{\sqrt{75r^3}}{\sqrt{108r}}$

$\frac{\sqrt{196q^5}}{\sqrt{484q}}$

$\frac{\sqrt{108p^5{q}^2}}{\sqrt{34p^3{q}^6}}$

$\frac{\sqrt{98r{s}^{10}}}{\sqrt{2r^3{s}^4}}$

$\frac{\sqrt{320m{n}^5}}{\sqrt{45m^7{n}^3}}$

$\frac{\sqrt{810c^3{d}^7}}{\sqrt{1000c^{5}d}}$

$\frac{\sqrt{98}}{14}$

$\frac{\sqrt{72}}{18}$

$\frac{5+\sqrt{125}}{15}$

$\frac{6-\sqrt{45}}{12}$

$\frac{\sqrt{96}}{\sqrt{150}}$

$\frac{\sqrt{28}}{\sqrt{63}}$

$\frac{\sqrt{26y^7}}{\sqrt{2y}}$

$\frac{\sqrt{15x^3}}{\sqrt{3x}}$

$\frac{10}{\sqrt{6}}$

$\frac{8}{\sqrt{3}}$

$\frac{6}{\sqrt{7}}$

$\frac{4}{\sqrt{5}}$

$\frac{3}{\sqrt{13}}$

$\frac{10}{\sqrt{11}}$

$\frac{10}{3\sqrt{10}}$

$\frac{2}{5\sqrt{2}}$

$\frac{4}{9\sqrt{5}}$

$\frac{9}{2\sqrt{7}}$

$-\frac{9}{2\sqrt{3}}$

$-\frac{8}{3\sqrt{6}}$

$\sqrt{\frac{3}{20}}$

$\sqrt{\frac{4}{27}}$

$\sqrt{\frac{7}{40}}$

$\sqrt{\frac{8}{45}}$

$\sqrt{\frac{19}{175}}$

$\sqrt{\frac{17}{192}}$

ⓐ $\frac{3}{3+\sqrt{11}}$ ⓑ $\frac{8}{1-\sqrt{5}}$

ⓐ $\frac{4}{4+\sqrt{7}}$ ⓑ $\frac{7}{2-\sqrt{6}}$

ⓐ $\frac{5}{5+\sqrt{6}}$ ⓑ $\frac{6}{3-\sqrt{7}}$

ⓐ $\frac{6}{6+\sqrt{5}}$ ⓑ $\frac{5}{4-\sqrt{11}}$

$\frac{\sqrt{3}}{\sqrt{m}-\sqrt{5}}$

$\frac{\sqrt{5}}{\sqrt{n}-\sqrt{7}}$

$\frac{\sqrt{2}}{\sqrt{x}-\sqrt{6}}$

$\frac{\sqrt{7}}{\sqrt{y}+\sqrt{3}}$

$\frac{\sqrt{r}+\sqrt{5}}{\sqrt{r}-\sqrt{5}}$

$\frac{\sqrt{s}-\sqrt{6}}{\sqrt{s}+\sqrt{6}}$

$\frac{\sqrt{150x^2{y}^6}}{\sqrt{6x^4{y}^2}}$

$\frac{\sqrt{80p^{3}q}}{\sqrt{5p{q}^5}}$

$\frac{15}{\sqrt{5}}$

$\frac{3}{5\sqrt{8}}$

$\sqrt{\frac{8}{54}}$

$\sqrt{\frac{12}{20}}$

$\frac{3}{5+\sqrt{5}}$

$\frac{20}{4-\sqrt{3}}$

$\frac{\sqrt{2}}{\sqrt{x}-\sqrt{3}}$

$\frac{\sqrt{5}}{\sqrt{y}-\sqrt{7}}$

$\frac{\sqrt{x}+\sqrt{8}}{\sqrt{x}-\sqrt{8}}$

$\frac{\sqrt{m}-\sqrt{3}}{\sqrt{m}+\sqrt{3}}$

A supply kit is dropped from an airplane flying at an altitude of 250 feet. Simplify $\sqrt{\frac{250}{16}}$ to determine how many seconds it takes for the supply kit to reach the ground.

A flare is dropped into the ocean from an airplane flying at an altitude of 1,200 feet. Simplify $\sqrt{\frac{1200}{16}}$ to determine how many seconds it takes for the flare to reach the ocean.

ⓐ Simplify $\sqrt{\frac{27}{3}}$ and explain all your steps.
ⓑ Simplify $\sqrt{\frac{27}{5}}$ and explain all your steps.
ⓒ Why are the two methods of simplifying square roots different?

1. ⓐ Approximate $\frac{1}{\sqrt{2}}$ by dividing $\frac{1}{1.414}$ using long division without a calculator.
2. ⓑ Rationalizing the denominator of $\frac{1}{\sqrt{2}}$ gives $\frac{\sqrt{2}}{2}.$ Approximate $\frac{\sqrt{2}}{2}$ by dividing $\frac{1.414}{2}$ using long division without a calculator.
3. ⓒ Do you agree that rationalizing the denominator makes calculations easier? Why or why not?