# Simplify a Radical

How to simplify a radical: 3 examples and their solutions.

## Example√4x^{2}

Change the factors of 4x^{2} to perfect squares.

4 = 2^{2}

x^{2} is already a perfect square.

So √4x^{2} = √2^{2}⋅x^{2}.

Take the squared factors, 2 and x,

out from the square root.

Then √2^{2}⋅x^{2} = 2x.

So 2x is the answer.

## Example√5a^{2}bc^{6}

Change the factors of 5a^{2}bc^{6} to perfect squares

as much as you can.

a^{2} is already a perfect square.

c^{6} = c^{3⋅2} = (c^{3})^{2}

Power of a Power

Take the squared factors, a and c^{3},

out from the square root.

And leave the non-squared factors, 5 and b,

in the square root.

So ac^{3}√5b is the answer.

## Example√12x^{9}

Change the coefficient 12 to its prime factorization:

12 = 2^{2}⋅3.

x^{9} is an odd power.

To make a perfect square,

split x^{9} to x^{8} and x.

x^{8} is an even power,

which will be a perfect square.

Change the factors of 2^{2}⋅3⋅x^{8}⋅x to perfect squares

as much as you can.

2^{2} is already a perfect square.

x^{8} = x^{4⋅2} = (x^{4})^{2}

Take the squared factors, 2 and x^{4},

out from the square root.

And leave the non-squared factors, 3 and x,

in the square root.

So 2x^{4}√3x is the answer.