Hello Kishor.

Most people ask this kind of question when dealing with very large numbers their calculators can't handle.

I'll show you with an example:

What is the value of 34.1^1722.3 ?

Clearly, many calculators would balk at this number. Another way to calculate it in order to express it in scientific notation would be to use logarithms.

The useful logarithm rule is this:

Given a = b^c, we can take the logarithm in any base d on both sides of this equation and get:

log_d a = log_d b^c = c log_d b.

Since scientific notation uses base ten (decimal), we want d = 10.

So for our example, we have:

Let a = 34.1^1722.3. Then

log₁₀ a = 1722.3 (log₁₀ 34.1) = 1722.3(1.532754379) = 2639.8629.

So a = 10^2639.8629. Now we break it up.

a = 10^2639.8629 = 10^{(0.8629 + 2639)} = 10^0.8629 × 10²⁶³⁹

a = 7.29 × 10²⁶³⁹.

That's the method I use. Here's the short form for n^m:

1. Calculate p = m log₁₀ n.
2. Break up p into its whole number (w) and its decimal (d) portions.
3. Calculate f = 10^d.
4. The answer expressed in scientific notation is n^m = f × 10^w.

Hope this helps,
Stephen La Rocque.