Estimating products of prime numbers

[ganesh.kumaraswamy]

There are some questions in the OG guide related to estimating the products of many numbers in terms of powers of 10. Their answer explanations aren't very useful, as it simply lists the math to arrive at the exact value, which isn't very practical on the test.

Since I can't present the actual text of the question(s), can I ask what's the best way to estimate the product of all prime numbers under, for example 20 or 30, in terms of a power of 10?

[RonPurewal]

There are some questions in the OG guide related to estimating the products of many numbers in terms of powers of 10. Their answer explanations aren't very useful, as it simply lists the math to arrive at the exact value, which isn't very practical on the test.

Since I can't present the actual text of the question(s), can I ask what's the best way to estimate the product of all prime numbers under, for example 20 or 30, in terms of a power of 10?

you could always estimate the numbers. if you're just asked for the closest power of 10, then you can use extremely crude estimates, since the different powers of 10 are, to say the least, REALLY far apart. it's hard to imagine how horrible your estimates would have to be to arrive at, say, 100,000 on a problem whose answer is supposed to be 1 million.

so, if you're finding the product of all the primes under twenty, that's 2x3x5x7x11x13x17x19.
i would just suck it up and multiply the first four numbers -- 2x3x5x7 = 10x21 = 210. let's estimate that as 200.
11 is about 10, 19 is about 20, so 11x19 is about 200.
the two remaining numbers are 13 and 17 -- just multiply them together (since they aren't terribly close to any "nice" numbers). you'll get 221. that's also "about 200" -- that's off by about 10%, but 10% is negligible compared to the difference between consecutive powers of 10.
so your number is, very approximately, 200 x 200 x 200 = 8,000,000, so the closest power of ten would be 10,000,000.