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Multiplying and dividing surds

It is easy to compute and simplify the product and division of surds.

$$\sqrt{2}\times\sqrt{8} = \sqrt{16} = 4,\quad\sqrt{\frac{18}{8}} = \sqrt{\frac{9}{4}} = \frac{3}{2}.$$

It uses the formula for the multiplication of square roots. $$\sqrt{x}\sqrt{y} = \sqrt{xy},\quad \frac{\sqrt{x}}{\sqrt{y}} = \sqrt{\frac{x}{y}}.$$

Let's illustrate the process to compute $\displaystyle\frac{\sqrt{18}}{\sqrt{10}}$

• Compute the prime factorisation of each number. $$18 = 2\times 3^2,\quad 50 = 2\times 5$$
• Simplify the product or the fraction. $$\frac{18}{10} = \frac{\Tblue{2}\times 3^2}{\Tblue{2}\times 5} = \frac{3^2}{5}$$
• Take the square root and take out numbers with even power. $$\sqrt{\frac{18}{10}} = \frac{\sqrt{\Tred{3^2}}}{\sqrt{5}} = \frac{\Tred{3}}{\sqrt{5}}$$
• Rationalise the denominator. $$\sqrt{\frac{18}{10}} = \frac{3}{\Tgreen{\sqrt{5}}} = \frac{3\times\Torange{\sqrt{5}}}{\Tgreen{\sqrt{5}}\times\Torange{\sqrt{5}}} = \frac{3\Torange{\sqrt{5}}}{5}$$