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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*}