Simplify: 6/(3+2sqrt3)

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To simplify the expression 6/(3+2√3), we can try rationalizing the denominator. Rationalizing the denominator means eliminating any square roots or radicals from the denominator by finding an equivalent expression without them.

Here are the steps to simplify the expression:

Step 1: Multiply the numerator and denominator by the conjugate of the denominator.

The conjugate of the denominator (3+2√3) is obtained by changing the sign of the square root term, so it becomes (3-2√3).

Multiplying the numerator and denominator by the conjugate, we get:

6/(3+2√3) * (3-2√3)/(3-2√3)

Step 2: Simplify the product of the fractions.

The numerator of the product is 6 * (3 - 2√3) = 18 - 12√3.

The denominator of the product is (3 + 2√3) * (3 - 2√3). This can be simplified using the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b). By applying this formula, we have:

(3 + 2√3) * (3 - 2√3) = 3^2 - (2√3)^2 = 9 - 4 * 3 = 9 - 12 = -3.

So, the denominator simplifies to -3.

Putting the simplified numerator and denominator together, we get:

(18 - 12√3)/(-3)

Step 3: Simplify the expression by factoring out any common factors.

In this case, all terms have a factor of 3. Factoring it out, we have:

(18/3 - 12√3/3)/(-3/3) = (6 - 4√3)/(-1)

Step 4: Simplify further if required.

Since the denominator is -1, we can multiply both the numerator and denominator by -1 to get a positive denominator:

(-1)((6 - 4√3)/(-1)) = (4√3 - 6)/1

So, the simplified expression is:

(4√3 - 6)/1 or simply 4√3 - 6.

Therefore, 6/(3+2√3) simplifies to 4√3 - 6.

Make sure to double-check the steps and the final answer, as there might be mistakes in the process. Good luck with your assignment!