root 3/1-cos30

To simplify the expression √(3/1-cos30), we can follow these steps:

Step 1: Simplify the expression inside the square root.
We know that cos(30) = √3/2. So, 1 - cos(30) = 1 - (√3/2). To simplify further, we need to rationalize the denominator by multiplying both the numerator and denominator by 2.

1 - cos(30) = 1 - (√3/2) * (2/2)
= 1 - (√3 * 2) / (2 * 2)
= 1 - (√6 / 4)
= (4 - √6) / 4

Step 2: Substitute the simplified expression back into the original square root expression.
√(3/1-cos30) becomes √(3/(4 - √6) / 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.

√3 / (4 - √6) / 4 = 2√3 / (4 - √6)

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 - √6) * (4 + √6) = 16 - (√6 * √6)
= 16 - 6
= 10

So, the denominator becomes 10.

Next, we substitute the simplified numerator and denominator back into the expression:

2√3 / (4 - √6) / 4 = 2√3 / (10/4)
= 2√3 * (4/10)
= 8√3 / 10
= (4√3) / 5

Therefore, the simplified form of √(3/1-cos30) is (4√3) / 5.