1. Like radical terms

A radical term has an index (root sign degree), a radical coefficient, and a radicand (the number under the root sign).

Example : in 4$\sqrt{2}$ $\rightarrow$ coefficient = 4 , index = 2 (square root) , radicand = 2.

Two radical terms are LIKE if they have the SAME index AND the SAME radicand.

• 5$\sqrt{2}$ and -3$\sqrt{2}$ are like (same index 2, same radicand 2).
• 5$\sqrt{2}$ and 5$\sqrt[3]{2}$ are NOT like (different indices).
• 7$\sqrt{2}$ and 7$\sqrt{3}$ are NOT like (different radicands).

2. Addition and Subtraction

We can only add/subtract like radical terms by adding/subtracting their coefficients.

Examples :

• $\sqrt{3}$ + 2$\sqrt{3}$ = (1 + 2)$\sqrt{3}$ = 3$\sqrt{3}$
• 3$\sqrt{5}$ - 4$\sqrt{5}$ + $\sqrt{5}$ = (3 - 4 + 1)$\sqrt{5}$ = 0$\sqrt{5}$ = 0
• 2$\sqrt{7}$ - 3$\sqrt{3}$ + $\sqrt{7}$ + 2$\sqrt{3}$ = (2 + 1)$\sqrt{7}$ + (-3 + 2)$\sqrt{3}$ = 3$\sqrt{7}$ - $\sqrt{3}$

3. Multiplication and Division

• Multiply coefficients $\times$ coefficients and radicands $\times$ radicands : $\sqrt{3}$ $\times$ $\sqrt{3}$ = $\sqrt{9}$ = 3.
• -5$\sqrt{2}$ $\times$ 3$\sqrt{2}$ = -15 $\times$ 2 = -30.
• 2$\sqrt{3}$ $\times$ (4$\sqrt{3}$ / $\sqrt{7}$)... follow the same rule : coefficients together, radicands together.
• The general rule : a$\sqrt{m}$ $\times$ b$\sqrt{n}$ = (a·b)$\sqrt{m·n}$
• For division : a$\sqrt{m}$ ÷ b$\sqrt{n}$ = (a/b)$\sqrt{m/n}$.

4. Distributing and expanding

• a(b + c) = ab + ac : 2($\sqrt{2}$ + $\sqrt{5}$) = 2$\sqrt{2}$ + 2$\sqrt{5}$.
• (x + y)(x - y) = x² - y² : ($\sqrt{2}$ + 1)($\sqrt{2}$ - 1) = 2 - 1 = 1.
• (x + y)² = x² + 2xy + y² : ($\sqrt{5}$ - 1)² = 5 - 2$\sqrt{5}$ + 1 = 6 - 2$\sqrt{5}$.
• (4 - 3$\sqrt{2}$)(4 + 3$\sqrt{2}$) = 16 - 18 = -2.

5. Making the denominator an integer (Rationalizing)

• Single radical denominator : multiply numerator and denominator by that radical.

Example : 6/$\sqrt{3}$ = (6$\sqrt{3}$)/($\sqrt{3}$·$\sqrt{3}$) = (6$\sqrt{3}$)/3 = 2$\sqrt{3}$.

• Sum/difference with a radical : multiply by the CONJUGATE.

($\sqrt{a}$ + $\sqrt{b}$) ↔ ($\sqrt{a}$ - $\sqrt{b}$) are conjugates; their product is a - b.

Example : 1/($\sqrt{3}$ + 2) = ($\sqrt{3}$ - 2)/(($\sqrt{3}$ + 2)($\sqrt{3}$ - 2)) = ($\sqrt{3}$ - 2)/(3 - 4) = ($\sqrt{3}$ - 2)/(-1) = 2 - $\sqrt{3}$.

Example : 20/(5 - $\sqrt{15}$) = 20(5 + $\sqrt{15}$)/(25 - 15) = 20(5 + $\sqrt{15}$)/10 = 2(5 + $\sqrt{15}$).

6. Useful area formulas

• Cube with edge s : area of one face = s² , lateral area = 4s² , total area = 6s² , volume = s³.
• Rectangular prism with dimensions l , w , h : total area = 2(lw + wh + hl) , volume = lwh.
• Square with side x : area = x². Rectangle : area = length $\times$ width.

7. WORKED EXAMPLES

Example 1 : Simplify 2$\sqrt{5}$ - 3$\sqrt{5}$ + $\sqrt{5}$.

• (2 - 3 + 1)$\sqrt{5}$ = 0. Answer : 0.

Example 2 : Find ($\sqrt{2}$ + 1)($\sqrt{2}$ - 1).

• = ($\sqrt{2}$)² - (1)² = 2 - 1 = 1.

Example 3 : If x = -4$\sqrt{3}$ and y = $\sqrt{3}$, find x $\times$ y.

• x $\times$ y = -4$\sqrt{3}$ $\times$ $\sqrt{3}$ = -4 $\times$ 3 = -12.

Example 4 : Find the area of a rectangle whose sides are (3$\sqrt{5}$ + 2) cm and (3$\sqrt{5}$ - 2) cm.

• Area = (3$\sqrt{5}$ + 2)(3$\sqrt{5}$ - 2) = (3$\sqrt{5}$)² - (2)² = 45 - 4 = 41 cm².