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$x=p_1p_2p_3$ ans $p_1+p_2+p_3 = p_4 ^ {p_5}$. If $5 \leq p_1, p_2, p_3,p_4,p_5 \leq 50$ and all of them are distinct primes then find the minimum value of $x$.

I tried with just plugging in primes and check if that works. But I don't think this will lead me to the minimum solution. How to systemically solve this?
Source: BdMO 2016 Chittagong Regional Secondary problem 8.

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    Maybe I'm wrong, but we have $p_4^{p_5}\ge 5^5=3125$ and $p_1+p_2+p_3<150$. – 2017-01-10
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    Does all primes need to be distinct? Also this question seems wrong since last three primes close to $50$ sum up to $131 < 5^5$ – 2017-01-10
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    @Xam The question is not saying $p_1 – 2017-01-10

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