SOLUTION: What is the approximate value of {{{ 3^sqrt(999) }}}?

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Question 1192841: What is the approximate value of +3%5Esqrt%28999%29+?
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
That's not too big to just punch into a scientific or graphing
calculator.  You get the answer immediately as
1.203242085 x 1015

That's no fun.

Your teacher should have given you one that will overflow any
calculator's capacity.  Like putting a couple more 9's under
that square root to make it too big for calculators:

matrix%282%2C1%2C%22%22%2C3%5Esqrt%2899999%29%29+

You can't punch that on a calculator.  It'll overflow.
So you'll have to use logarithms.  Set x equal to the quantity

x%22%22=%22%22%29matrix%282%2C1%2C%22%22%2C3%5Esqrt%2899999%29%29+

Use your calculator to get the exponent sqrt%2899999%29=316.2261849

x%22%22=%22%22%29matrix%282%2C1%2C%22%22%2C3%5E316.2261849%29

Take the log (base 10) of both sides:

log%28%28x%29%29%22%22=%22%22matrix%282%2C1%2C%22%22%2Clog%28%283%5E316.2261849%29%29%29

log%28%28x%29%29%22%22=%22%22316.2261849%2Alog%28%283%29%29

Use your calculator to get the right side:

log%28%28x%29%29%22%22=%22%22150.8782341

Raise 10 to both sides' power:

10%5Elog%28%28x%29%29%22%22=%22%2210%5E150.8782341

The left side is just x.  The right side is too big for 
calculators, so split the exponent into an integer and
a decimal

x%22%22=%22%2210%5E%28150%2B0.8782341%29

Write the right side as the product of powers of 10

x%22%22=%22%2210%5E150%2A10%5E0.8782341%29

Next use your calculator to get 10%5E0.8782341=7.554993627

x%22%22=%22%2210%5E150%2A7.554993627%29

x%22%22=%22%227.554993627%2A10%5E150%29

That's the answer in scientific notation to the problem
I made up.  But that's the way to do them if they're too
big for a calculator.

Edwin