Simplifying Square Roots

(1) Simplifying Square Roots Worksheets

(2) Perfect Square Method

(3) Prime Factorization Method

(4) Multiplying Square Roots

(5) Square Roots Involving Fractions

(6) Other Videos on Square Roots

This lens is going to discuss two specific ways of simplifying basic square roots. The first is the perfect square method.

The Perfect Square Method

The goal of this method is to factor the number inside the square root into a product of perfect squares and one non-perfect square.

Perfect Squares

A perfect square is any integer such that can be written as the product of another integer and itself. For example, 25 is a perfect square because it is equal to 5 times 5.

For our purposes we are going to focus on five specific perfect squares:

4 = 22
9 = 32
25 = 52
49 = 72
121 = 112

If you notice this is essentially the first 5 prime numbers squared. We don't worry about 16 or 36 or 64 because they themselves can be factored into perfect squares.

Procedure

1) Factor into perfect squares - The easiest way to do this is to try each of the five. For example, 72, we first see if 4 goes into 72. It does so we rewrite 72 as 4*18. Now we look at 18, 4 doesn't go into 18. However, 9 does so we rewrite it as 72 = 4*9*2. Obviously 2 is not a perfect square so we are done.

2) Remove perfect squares from square root - The product rule for square roots allows you to rewrite a square root as a product of square roots.

3) Simplify - Simplify the square roots of the perfect squares and remultiply.

The following video gives a brief explanation of the above and then the videos after that are a few examples using the perfect square method.

Check out this Square Root Calculator to help keep you on the right track!

Above we discussed the Perfect Square Method for simplifying square roots. This sections is on simplifying square roots using the prime factorization method. The goal of this method is to factor the number into prime factors.

Procedure

1) Factor - The first step is to factor the number into prime factors. There are different methods of doing this. The most common method is a factor tree.

2) Pair Up the Primes - Go through the prime factors and pair them up. After the pairs you then list the unpaired factors.

3) Remove the Pairs - Remove each pair from the square root and place ONE representative of the pair outside the square root.

4) Remultiply - The last step is to remultiply inside and outside the square root.

The following video is a tutorial on how to do this. The videos after that are examples or how to apply this method.

Multiplying square roots is a pretty simple concept. Essentially what you need to do is multiply the numbers outside square root symbols together and then multiply the numbers inside the square roots together. To see examples of this view the next two videos.

There are two important points to make about fractions in square roots.

1) If a fraction is inside a square root it can be seperated into a square root over a square root.

2) It is not acceptable to leave a square root in the denominator. To avoid this you have to multiply the numerator and denominator by the square root that is in the denominator.

The next couple of videos show how to deal with the fractions in square roots.