Rationalize the denominator and write your answer in lowest terms

(2 √ 6 + √ 3)/ ( √ 6- √ 3)

multiply top and bottom by √6 + √3

(2 √ 6 + √ 3)/ ( √ 6- √ 3) * (√6 + √3)/(√6 + √3)
= ( 12 + 3√18 + 3)/(6-3)
= (15 + 9√3)/3
= 5 + 3√2

5+3√2

To rationalize the denominator, we need to get rid of any square roots in the denominator.

To do this, we multiply both the numerator and the denominator by the conjugate of the denominator, which in this case is (√6 + √3).

When multiplying the numerator and the denominator by (√6 + √3), we need to apply the distributive property.

Let's go through each step to rationalize the denominator:

Step 1: Multiply the numerator and the denominator by (√6 + √3):

((2√6 + √3) * (√6 + √3)) / ((√6 - √3) * (√6 + √3))

Step 2: Apply the distributive property to both the numerator and the denominator:

(2√6 * √6 + 2√6 * √3 + √3 * √6 + √3 * √3) / (√6 * √6 - √6 * √3 + √3 * √6 - √3 * √3)

Simplifying this expression will help us get closer to the answer.

Step 3: Simplify the numerator:

(2 * 6 + 2√18 + √18 + 3) / (6 - 3)

Step 4: Combine like terms:

(12 + 3√18 + √18 + 3) / 3

Step 5: Simplify the numerator further:

(15 + 4√18) / 3

Step 6: Simplify the square root:

(15 + 4√(9 * 2)) / 3

(15 + 4√9 * √2) / 3

(15 + 4 * 3 * √2) / 3

(15 + 12√2) / 3

Step 7: Divide the numerator by the denominator:

(15/3 + 12√2/3)

5 + 4√2

Therefore, the rationalized form of the expression is 5 + 4√2.