Factoring radicals

When factoring radicals, the following formula is useful, important, and a must to simplify radicals:

√(a × b) = √(a) × √(b)

The formula tells you that the square root of the multiplications is the same thing as the multiplication of the square roots.

Let's not spend too much time on definition and English here. I am no English major. Let's start with examples.

Once you know how to factor radicals, simplifying radicals(the ultimate goal) is very straightforward.

Example #1

Factor √(24)

24 = 12 × 2

24 = 4 × 6

24 = 3 × 8

Therefore, √(24) can be factored 3 different ways. Using the formula above, we get:

√(24) = √(2 × 12) = √(2) × √(12)

√(24) = √(4 × 6) = √(4) × √(6)

√(24) = √(3 × 8) = √(3) × √(8)

However, if you want to simplify, the trick is to factor 24 so that one, two, or all factors are perfect squares so you can take the square roots of those factors to get whole numbers.

Since √(4) = 2, √(24) = √(4 × 6) = √(4) × √(6)= 2 × √(6)

If you try to simplify using the equation 24 = 2 × 12, you will get the same answer as above.

√(24)= √(12 × 2)= √(4 × 3 × 2) = √( 4 × 6) = √(4) × √(6)= 2 × √(6)

Example #2

Factor and simplify √(72)

72 = 9 × 8

72 = 36 × 2

72 = 3 × 24

72 = 18 × 4

72 = 12 × 6

A lot more choices than before! So factoring radicals can quickly become a lot of work.

√(72) = √(9 × 8) = √(9) × √(8)

√(72) = √(36 × 2) = √(36) × √(2)

√(72) = √(3 × 24) = √(3) × √(24)

√(72) = √(18 × 4) = √(18) × √(4)

√(72) = √(12 × 6) = √(12) × √(6)

To simplify now, we can use √(72) = √(9 × 8) = √(9) × √(8)

√(72) = √(9 × 8) = √(9) × √(8) = 3 × √ (8) = 3 × √ (4 × 2) = 3 × √(4) × √(2) = 3 × 2 × √ (2) = 6 × √ (2)

The problem can be simplified faster using √(72) = √(36 × 2) = √(36) × √(2) = 6 × √ (2)

Conclusion: When simplifying and factoring radicals, carefully choose your factors for faster results

Did you struggle to understand this lesson about factoring radicals? Review some important basic algebra concepts

Algebra ebook

Recent Articles

  1. Irrational Root Theorem - Definition and Examples

    Dec 01, 21 04:17 AM

    What is the irrational root theorem? Definition, explanation, and easy to follow examples.

    Read More

Enjoy this page? Please pay it forward. Here's how...

Would you prefer to share this page with others by linking to it?

  1. Click on the HTML link code below.
  2. Copy and paste it, adding a note of your own, into your blog, a Web page, forums, a blog comment, your Facebook account, or anywhere that someone would find this page valuable.