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My 12-year-old son had a math competition question:

If 10^x = 50, find 2500^(1/x)

I believe the answer is 100, but I had to use a calculator to solve it.  Calculators were not allowed during the test.  So, can you show the steps in solving this without using a calculator?

Thanks

Questioner:Jimmy
Country:Alabama, United States
Private:No

Subject:Math competition question

Question:
My 12-year-old son had a math competition question:

If 10^x = 50, find 2500^(1/x)

I believe the answer is 100, but I had to use a calculator to solve it.  Calculators were not allowed during the test.  So, can you show the steps in solving this without using a calculator?

Thanks

If 10^x = 50,  x = log 50 = 1 + log 5

If y = 2500^(1/x),

log y = 1/x log(2500)

log y = 1/x ( log(25) +  log(100))

log y = 1/x (2(log 5 +  1))

log y = 1/x (2x) = 2

y = 10^2 = 100

Oh, I forgot -- you said your son is 12; he is not allowed to use logarithms.  (They are for ages 15 to adult -- remember all those games you bought for the kids?)

So let's try this -- remember that if you raise a power to a power, keep the base, multiply exponents.
Like  (10^A)^B = 10^(AB)

Problem:  If 10^x = 50, find 2500^(1/x)

2500 = 50^2, SO

2500^(1/x) = (50^2)^(1/x) = 50^(2/x)

But 50 = 10^x, so

50^(2/x) = (10^x)^(2/x) =

(10^2x/x) = 10^2 = 100

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