Factoring radicals

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

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)
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

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