Simplifying a radical is sometimes called reducing a radical.  It is not the same thing as estimating a radical.  Simplifying a radical gives the radical in its simplest form and represents the exact same value.

It is helpful to know all the perfect squares to at least 100 when simplifying radicals:

1² = 1; 2² = 4; 3² = 9; 4² = 16; 5² = 25; 6² = 36; 7² = 49; 8² = 64; 9² = 81; 10² = 100

We will use the law of radicals that states:  rad(a*b) = rad(a) * rad(b)

That is, if we are taking a radical of a product of two numbers, that is the same as multiplying the radical of each factor.

Suppose we want to simplify SQRT(12).  12 is not a perfect square.  We find the BIGGEST perfect square that is a factor of 12.  The biggest perfect square that is a factor of 12 is 4, since 4*3 = 12.

We re-write:  SQRT(12) = SQRT(4*3)

We use the law of radicals to change SQRT(4*3) to SQRT(4)*SQRT(3).  We know that SQRT(4) = 2.

So...

SQRT(12) =

SQRT(4*3) =

SQRT(4)*SQRT(3) =

2*SQRT(3)  (2 times the square root of 3).

We can simplify algebraic radicals as well.  When working with square roots (radicals with root 2), radicands with even square roots are perfect squares.  For example x², x⁴, x⁶, x⁸ are all perfect squares (with positive roots x, x², x³, and x⁴ respectively).  (Review the laws of exponents to convince yourself this is true).  If we are given a variable with an odd number exponent, re-write it as a product of an even number exponent.

Examples - Simplify:

SQRT(x⁴) = x²

SQRT(x⁹) = SQRT(x⁸*x) = SQRT(x⁸)*SQRT(x) = x⁴SQRT(x)

Another consideration when simplifying radicals is that we cannot have a radical in the denominator of a fraction; it is not simplified until we "get rid of" the radical.  We do this by rationalizing the denominator.

These class notes provide further explanation and examples of simplifying radicals.