To simplify the expression β(3/1-cos30), we can follow these steps:
Step 1: Simplify the expression inside the square root.
We know that cos(30) = β3/2. So, 1 - cos(30) = 1 - (β3/2). To simplify further, we need to rationalize the denominator by multiplying both the numerator and denominator by 2.
1 - cos(30) = 1 - (β3/2) * (2/2)
= 1 - (β3 * 2) / (2 * 2)
= 1 - (β6 / 4)
= (4 - β6) / 4
Step 2: Substitute the simplified expression back into the original square root expression.
β(3/1-cos30) becomes β(3/(4 - β6) / 4).
Step 3: Simplify further.
To simplify the square root of a fraction, we can simplify the numerator and denominator separately.
Let's simplify the numerator first.
β3 / (4 - β6) / 4 = 2β3 / (4 - β6)
To simplify the denominator, we need to rationalize it, just like we did in Step 1. We multiply both numerator and denominator by the conjugate of the denominator.
(4 - β6) * (4 + β6) = 16 - (β6 * β6)
= 16 - 6
= 10
So, the denominator becomes 10.
Next, we substitute the simplified numerator and denominator back into the expression:
2β3 / (4 - β6) / 4 = 2β3 / (10/4)
= 2β3 * (4/10)
= 8β3 / 10
= (4β3) / 5
Therefore, the simplified form of β(3/1-cos30) is (4β3) / 5.