### Instruction: The sum $10 e^{2 \pi i/11} + 10 e^{15 \pi i/22}$ is expressed as $re^{i \theta}.$  Enter the ordered pair $(r, \theta).$.### Response: I want to find $r$ and $\theta$ such that $10 e^{2 \pi i/11} + 10 e^{15 \pi i/22} = r e^{i \theta}.$
To do that, I need to write the left-hand side in the form of a single complex exponential.
I notice that both terms have a common factor of $10$, so I can factor that out and get $10 (e^{2 \pi i/11} + e^{15 \pi i/22}).$
Now I need to find a way to combine the two exponents inside the parentheses.
I see that they have different denominators, so I want to find a common denominator for them.
The lowest common denominator is 22, so I multiply the first exponent by 2 and the second exponent by 1 to get $10 (e^{4 \pi i/22} + e^{15 \pi i/22}).$
Now I can use the property of complex exponents that says $e^{a} + e^{b} = e^{(a + b)/2} (e^{(a - b)/2} + e^{(b - a)/2}).$
Applying this property to the two exponents, I get $10 e^{(4 \pi i/22 + 15 \pi i/22)/2} (e^{(4 \pi i/22 - 15 \pi i/22)/2} + e^{(15 \pi i/22 - 4 \pi i/22)/2}).$
Simplifying the fractions, I get $10 e^{19 \pi i/44} (e^{-11 \pi i/44} + e^{11 \pi i/44}).$
Now I recognize that the second factor is the sum of two complex conjugates, which is always a real number.
In fact, it is the same as $2 \cos(11 \pi / 44)$, by using the formula for cosine in terms of complex exponentials.
So I can rewrite the expression as $20 \cos(11 \pi / 44) e^{19 \pi i/44}.$
This is now in the form of $re^{i \theta}$, where $r = 20 \cos(11 \pi / 44)$ and $\theta = 19 \pi / 44.$