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# Simplifying surds

We generally simplify surds as the product of positive integers times the surd of the product of prime numbers.

$\sqrt{9}$ simplifies to $3$ and $\sqrt{8}$ simplifies to $2\sqrt{2}$.

We illustrate the simplification process with $\sqrt{24}$.

• Find the prime factorisation of the number $$24 = 2^3\times 3$$
• Put aside the even powers of the prime numbers. $$24 = \Tblue{2^2}\times \Tgreen{2}\times \Tgreen{3}$$
• Take the square root of the even powers by dividing the power by 2. Multiply the numbers with multiplicity 1 and leave them as a surd. $$\sqrt{24} = \sqrt{\Tblue{2^2}}\times\sqrt{\Tgreen{2\times 3}} = \Tblue{2}\sqrt{\Tgreen{6}}$$

Here are other simplified surds.

\begin{align*} &\qquad\sqrt{4} = \Tblue{2},\quad \sqrt{75} = \sqrt{5^3} = \sqrt{\Tblue{5^2}\times \Tgreen{5}} = \Tblue{5}\sqrt{\Tgreen{5}},\\ &\sqrt{72} = \sqrt{2^3\times 3^2} = \sqrt{\Tblue{2^2}\times \Tgreen{2}\Tblue{\times 3^2}} = \Tblue{2\times 3}\sqrt{\Tgreen{2}} = \Tblue{6}\sqrt{\Tgreen{2}}. \end{align*}