help please dividing surds

Question

Can anyone explain how I simplify $$5\frac{\sqrt{44}}{\sqrt{66}}$$

thanks

Answer 1

You shouldn't have a surd as a denominator so you rationalise the denominator by multiplying both top and bottom by $\sqrt{66}$ .

Answer 2

$$\frac{3\sqrt{22}}{\sqrt{66}}=\frac{3\sqrt{22}\sqrt{66}}{\sqrt{66}\sqrt{66}}=\frac{3\sqrt{22*66}}{66}=\frac{3\sqrt{ 22*22*3}}{66}=\frac{3* 22\sqrt{3}}{66}=\sqrt{3}$$

Answer 3

METHOD 1: FRACTION OF SQUARE ROOTS

you have $$3\frac{\sqrt{22}}{\sqrt{66}} = 3\sqrt{\frac{22}{66}}=3\sqrt{\frac{2\times 11}{2\times 3\times 11}}=3\sqrt{\frac{2}{2}\times \frac{1}{3}\times\frac{11}{11}} = 3\sqrt{1\times \frac{1}{3}\times 1}=3\sqrt{\frac{1}{3}}=3\frac{1}{\sqrt{3}}=\frac{3}{\sqrt{3}}=\frac{\sqrt{3}\sqrt{3}}{\sqrt{3}}=\sqrt{3}$$

METHOD 2: MULTIPLY TOP AND BOTTOM

Notice that $1 = \frac{\sqrt{66}}{\sqrt{66}}$. Hence $$3\frac{\sqrt{22}}{\sqrt{66}} = 3\frac{\sqrt{22}}{\sqrt{66}}1 = 3\frac{\sqrt{22}}{\sqrt{66}}\frac{\sqrt{66}}{\sqrt{66}}=3\frac{\sqrt{22\times 66}}{\sqrt{66}^2}=3\frac{\sqrt{2\times 11\times 6\times 11}}{66}= 3\frac{\sqrt{2\times 6\times 11\times 11}}{66}=3\frac{\sqrt{2\times 2 \times 3 \times 121}}{66}=3\frac{\sqrt{4\times 6\times 121}}{66}=3\frac{2\times 11 \times\sqrt{3}}{66}=3\frac{22 \times\sqrt{3}}{66}=3\frac{ \sqrt{3}}{3}=\sqrt{3}$$

METHOD 3: SPLIT

$$3\frac{\sqrt{22}}{\sqrt{66}} = 3\frac{\sqrt{2\times 11}}{\sqrt{6\times 11}}=3\frac{\sqrt{2}\times\sqrt{11}}{\sqrt{6}\times \sqrt{11}}=3\frac{\sqrt{2}}{\sqrt{6}}\times\sqrt{\frac{11}{11}}=3\frac{\sqrt{2}}{\sqrt{6}}\times1=3\frac{\sqrt{2}}{\sqrt{6}} = 3\frac{\sqrt{2}}{\sqrt{2\times 3}} = 3\frac{\sqrt{2}}{\sqrt{2}\times\sqrt{3}} = 3\frac{\sqrt{2}}{\sqrt{2}}\frac{1}{\sqrt{3}}=3\sqrt{\frac{2}{2}}\frac{1}{\sqrt{3}}=3\times 1\times\frac{1}{\sqrt{3}}=3\frac{1}{\sqrt{3}} = \frac{3}{\sqrt{3}} =\frac{\sqrt{3}^2}{\sqrt{3}} = \frac{\sqrt{3}\times\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{3}}{\sqrt{3}}\sqrt{3}=\sqrt{\frac{3}{3}}\sqrt{3}=\sqrt{1}\times\sqrt{3}=1\times\sqrt{3}=\sqrt{3}$$