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
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