You have two conditions, the letter you draw must be E and the number you roll must be even, and both of those conditions have to happen at the same time (aka they're dependent on each other). Because of that, you have to multiply the probability you'll get the first condition by the probability you get the second one.
You have 5 vowels, so the probability of drawing E, one of the vowels, is 1/5.
You have 6 numbers on your standard die. Three of the numbers, 2, 4, 6 are even. The probability of rolling an even number is 3/6 = 1/2.
Multiply these two probabilities together to get the probability of both of those happening at the same time:
(1/5)*(1/2) = 1/10 = 0.100
