1. The golden rule: NO distribution over + and -

$\sqrt{a ± b}$ $\neq$ $\sqrt{a}$ ± $\sqrt{b}$ and $\sqrt[3]{a ± b}$ $\neq$ $\sqrt[3]{a}$ ± $\sqrt[3]{b}$

• $\sqrt{2}$ + $\sqrt{4}$ $\neq$ $\sqrt{6}$ ($\sqrt{2}$ + $\sqrt{4}$ = $\sqrt{2}$ + 2)
• $\sqrt[3]{2}$ + $\sqrt[3]{3}$ $\neq$ $\sqrt[3]{5}$
• $\sqrt{5}$ - $\sqrt{2}$ $\neq$ $\sqrt{3}$
• $\sqrt[3]{7}$ - $\sqrt[3]{3}$ $\neq$ $\sqrt[3]{4}$

We can only add/subtract like radical terms (same index + same radicand).

2. Simplifying a square root

Write the radicand as (perfect square) $\times$ (rest), then take the square root of the perfect square out.

• $\sqrt{32}$ = $\sqrt{16 $\times$ 2}$ = $\sqrt{16}$ $\times$ $\sqrt{2}$ = 4$\sqrt{2}$
• $\sqrt{50}$ = $\sqrt{25 $\times$ 2}$ = 5$\sqrt{2}$
• $\sqrt{98}$ = $\sqrt{49 $\times$ 2}$ = 7$\sqrt{2}$ , $\sqrt{8}$ = 2$\sqrt{2}$ , $\sqrt{18}$ = 3$\sqrt{2}$

3. Simplifying a cube root

Write the radicand as (perfect cube) $\times$ (rest).

• $\sqrt[3]{24}$ = $\sqrt[3]{8 $\times$ 3}$ = 2$\sqrt[3]{3}$
• 2$\sqrt[3]{16}$ = 2$\sqrt[3]{8 $\times$ 2}$ = 2 $\times$ 2$\sqrt[3]{2}$ = 4$\sqrt[3]{2}$
• $\sqrt[3]{128}$ = $\sqrt[3]{64 $\times$ 2}$ = 4$\sqrt[3]{2}$
• ∛-2 ... $\sqrt[3]{54}$ = $\sqrt[3]{27 $\times$ 2}$ = 3$\sqrt[3]{2}$

4. Combining like radicals

Simplify each term first, then add/subtract like terms.

Example : $\sqrt{50}$ + $\sqrt{8}$ = 5$\sqrt{2}$ + 2$\sqrt{2}$ = 7$\sqrt{2}$.

Example : 2$\sqrt{3}$ + 5$\sqrt{27}$ - $\sqrt{48}$ = 2$\sqrt{3}$ + 5(3$\sqrt{3}$) - 4$\sqrt{3}$ = 2$\sqrt{3}$ + 15$\sqrt{3}$ - 4$\sqrt{3}$ = 13$\sqrt{3}$.

Example : $\sqrt[3]{16}$ - $\sqrt[3]{2}$ = 2$\sqrt[3]{2}$ - $\sqrt[3]{2}$ = $\sqrt[3]{2}$.

5. Important identities (Remember)

• (a + b)(a - b) = a² - b²
• (a ± b)² = a² ± 2ab + b²
• $\sqrt{m/n}$ = $\sqrt{m}$ / $\sqrt{n}$ and $\sqrt[3]{m/n}$ = $\sqrt[3]{m}$ / $\sqrt[3]{n}$

6. WORKED EXAMPLES

Example 1 : Simplify 2$\sqrt{32}$ - $\sqrt{18}$ + $\sqrt{50}$.

• 2$\sqrt{32}$ = 2(4$\sqrt{2}$) = 8$\sqrt{2}$ ; $\sqrt{18}$ = 3$\sqrt{2}$ ; $\sqrt{50}$ = 5$\sqrt{2}$
• Total = (8 - 3 + 5)$\sqrt{2}$ = 10$\sqrt{2}$.

Example 2 : Simplify $\sqrt[3]{192}$ - 3$\sqrt[3]{375}$ + 2$\sqrt[3]{24}$.

• $\sqrt[3]{192}$ = $\sqrt[3]{64$\times$3}$ = 4$\sqrt[3]{3}$ ; 3$\sqrt[3]{375}$ = 3$\sqrt[3]{125$\times$3}$ = 15$\sqrt[3]{3}$ ; 2$\sqrt[3]{24}$ = 2$\sqrt[3]{8$\times$3}$ = 4$\sqrt[3]{3}$
• Total = (4 - 15 + 4)$\sqrt[3]{3}$ = -7$\sqrt[3]{3}$.

Example 3 : If the perimeter of a rectangle is $\sqrt{300}$ meters and its length is $\sqrt{27}$ meters, find its width.

• Perimeter = 2(length + width) $\rightarrow$ $\sqrt{300}$ = 2($\sqrt{27}$ + width)
• $\sqrt{300}$ = 10$\sqrt{3}$ , $\sqrt{27}$ = 3$\sqrt{3}$ $\rightarrow$ 10$\sqrt{3}$ = 6$\sqrt{3}$ + 2·width $\rightarrow$ 2·width = 4$\sqrt{3}$ $\rightarrow$ width = 2$\sqrt{3}$ m.

Example 4 : A square has area 144 m², find the length of its diagonal.

• Side = $\sqrt{144}$ = 12 m. Diagonal d satisfies d² = 12² + 12² = 288 $\rightarrow$ d = $\sqrt{288}$ = 12$\sqrt{2}$ m.