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What is the value of $n$ such that $10^n = 10^{-6}\times \sqrt{\frac{10^{46}}{0.0001}}$

 Jul 23, 2019
 #1
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Solve for n:
10^n = sqrt(10^46/0.0001)/10^6

sqrt(10^46/0.0001)/10^6 = sqrt(10^46/(1/10000))/1000000:
10^n = sqrt(10^46/(1/10000))/1000000

10000000000000000000 = 2^19·5^19:
2^n·5^n = 2^19·5^19

Equate exponents of 2 and 5 on both sides:
n = 19 and n = 19

All equations give n = 19 as the solution:
n = 19

 Jul 23, 2019
 #2
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10^n = sqrt(10^46/0.0001)/10^6

        = sqrt(10^50)  / 10^6

        = 10^25  / 10^6

        = 10^19                     n = 19

 Jul 23, 2019

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