Next, we simplify the numerator: 17 + 2(30)^1/2 = 17 + 2√30
Now we substitute both parts back into the original expression: (17 + 2√30) / (√15 + 2)
Now we need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator: [(17 + 2√30) / (√15 + 2)] * [(√15 - 2) / (√15 - 2)]
Expanding out the numerator and denominator gives us: [(17√15 + 2√30 - 34) / (15 - 4)]
Let's break down the expression step by step:
First, we simplify the denominator:
√15 + √2 = √15 + √(4 * 1) = √15 + 2√1 = √15 + 2
Next, we simplify the numerator:
17 + 2(30)^1/2 = 17 + 2√30
Now we substitute both parts back into the original expression:
(17 + 2√30) / (√15 + 2)
Now we need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator:
[(17 + 2√30) / (√15 + 2)] * [(√15 - 2) / (√15 - 2)]
Expanding out the numerator and denominator gives us:
[(17√15 + 2√30 - 34) / (15 - 4)]
Simplifying further, we get:
(17√15 + 2√30 - 34) / 11
Therefore, the final simplified expression is:
(17√15 + 2√30 - 34) / 11