A root is the opposite of an exponent.

The most common root you’ll see is the square root.

sqrt{9} = 3

Why? Because:

3^2 = 9

So when we simplify a root, we’re rewriting it in the simplest form possible, using smaller numbers and no perfect squares left inside the root.

What is a radical?

The symbol for sqrt{} ​ is called a radical.

Example:

sqrt{20}​

This means:
“What number squared gives 20?”
Since 20 is not a perfect square, we simplify it.

Perfect squares (VERY important)

Perfect squares are numbers you get by squaring whole numbers:

Memorizing these helps a lot.

How to simplify square roots (step-by-step)

Example: sqrt{20}​

1. Break 20 into factors:

   $$4\times 5$$

2. Rewrite the root:

   sqrt{20} = 4×5​

3. Take the square root of the perfect square:

   sqrt{4} = 2

4. Final answer:

   2sqrt{5}​

✔ This is simplified because no perfect square is left inside the radical.

Roots with variables

Example: sqrt{x^2}​

sqrt{x^2} = x

Because:

x2→x

Example: sqrt{12x^2}​

1. Factor:

   12x^2 = 4 × 3 × x^2

2. Take out perfect squares:

   sqrt{4} = 2, sqrt{x^2} = x

3. Final answer:

   2xsqrt{3}​

Important rule to remember

sqrt{ab} = sqrt{a} sqrt{b}​

This rule is what allows us to split numbers apart and simplify.