A radical is a root, usually a square root.

Example:

sqrt{9} = 3

The radical symbol has parts:

sqrt{^[3] {8}}​

• 3 → index (what root it is)

• 8 → radicand (the number inside)

• If no index is written, it’s a square root

Radical notation (how roots are written)

Square root

sqrt{16} = 4

Cube root

sqrt{^[3] {27}} = 3

With variables

sqrt{x^2} = x

Why radical laws exist

Radical laws let us:

• simplify expressions

• multiply and divide radicals

• solve equations involving roots

They work because roots are just fractional exponents.

The main radical laws (with examples)

1. Product Rule for Radicals

sqrt{a} * sqrt{b} = sqrt{ab}​

Example:

sqrt{2} * sqrt{8} = sqrt{16} = 4

2. Quotient Rule for Radicals

sqrt{a} / sqrt{b} = sqrt{{a} / {b}} (b ≠ 0)

Example:

sqrt{18} / sqrt{2} = sqrt{9} = 3

3. Simplifying Radicals (perfect squares come out)

sqrt{a^2 b} = a*sqrt{b} 

Example:

20 = sqrt{4 * 5} = 2sqrt{5}​

4. Radical and Exponent Connection

sqrt{a} = a^(1/2)​

Example:

sqrt{9} = 9^{1/2} = 3

This becomes very useful later in algebra.

5. Like Radicals Can Be Combined

You can add or subtract radicals only if they have the same radicand.

✔ 3sqrt{5} + 2sqrt{5} = 5sqrt{5}​

❌ sqrt{3} + sqrt{5}​ (cannot be combined)

Important things radicals CANNOT do

• ❌ sqrt{a+b} ≠ sqrt{a} + sqrt{b}​

• ❌ You can’t add or subtract unlike radicals