In this lesson, the term "radical" will refer to "square root".

You have dealt with the square root of a perfect square and the cube root of a perfect cube.
You have also dealt with finding decimal approximation (estimations) of radicals of non-perfect squares, which yielded irrational numbers.

In Algebra, we are going to now deal with the process of simplifying radicals.
This is NOT a decimal estimation. These results will still have a radical in the answer.

To simplify a radical means to write it in another "form"
which is equivalent to the given radical.

This "new form" will give a better indication of the size of the radical, and will make it
easier to deal with calculations involving the radical.

 Simplifying Square Roots:

 A square root is in simplest form when 1. the radicand contains no perfect square factors 2. the radicand is not a fraction 3. there are no radicals in the denominator of a fraction.

Find the largest perfect square factor (the largest perfect square that divides into 48 with no remainder). You need to be familiar with a list of perfect squares.

Give each factor its own radical sign.

Reduce the "perfect square" radical that was created.

 Don't worry if you do not pick the LARGEST perfect square factor to start. You can still get the correct answer, but you will have to repeat the process. See what happens if we choose 4 instead of 16 to start:          Notice how the out-front 2 in the second line is multiplied along for the rest of the problem.

The number 23 cannot be factored by any of the perfect squares (23 is prime). This is a trick question as it is already in simplest form and cannot be reduced further.

The problem has a number in front of the radical. This number is being multiplied times the radical, and that multiplication will carry through in the solution. The radical is simplified and the six is multiplied times that simplification.

A full explanation of dealing with this type of question can be fournd at: Divide Radicals & Rationalize Denominators.

If there is a
fraction under the radical symbol, the radical is NOT in simplest form.

Separate the problem by giving the numerator and denominator their own radical symbols. This is an application of the Quotient Rule (seen at the right).

While this solution may be better, it is still not in simplest form, since it leaves a

Now, let's get rid of the radical in the denominator by multiplying by a fraction whose value is "one".

This process of removing a radical from the denominator is referred to as "rationalizing the denominator" because it turns the denominator into a rational (not irrational) value.

Extra: Let's take a quick peek at how to rationalize cube roots:

 Simplifying Cube Roots:

Find the largest perfect cube factor (the largest perfect cube that divides into 24 with no remainder).

Give each factor its own radical symbol.

Reduce the "perfect cube" radical that was created.

 Perfect Squares 0 = 0 x 0 1 = 1 x 1 4 = 2 x 2 9 = 3 x 3 16 = 4 x 4 25 = 5 x 5 36 = 6 x 6 49 = 7 x 7 64 = 8 x 8 81 = 9 x 9 100 = 10 x 10 121 = 11 x 11 144 = 12 x 12 169 = 13 x 13 196 = 14 x 14 225 = 15 x 15

 Square Roots

 Prime Factorization Radicals can also be simplified by expressing the radicand using prime factorization and looking for groups of two similar factors to form a perfect square factor.

 Product Rule where a ≥ 0, b≥ 0 "The square root of a product is equal to the product of the square roots of each factor." This theorem allows us to use this method of simplifying radicals.

 Quotient Rule where a ≥ 0, b > 0 "The square root of a quotient is equal to the quotient of the square roots of the numerator and denominator."

 Perfect Cubes 0 = 0 x 0 x 0 1 = 1 x 1 x 1 8 = 2 x 2 x 2 27 = 3 x 3 x 3 64 = 4 x 4 x 4 125 = 5 x 5 x 5 216 = 6 x 6 x 6 243 = 7 x 7 x 7 512 = 8 x 8 x 8 729 = 9 x 9 x 9

 Cube Roots

 When working with radicals, the term "simplify" means to find an equivalent expression.

It does not mean to find a decimal approximation.