Math CentralQuandaries & Queries


Question from Kira, a student:

Is there an easy way to calculate a number with a large exponent? For example, 2(10)^35.


We have two responses for you

Ten to the power of n is a 1 followed by n zeros. For example, 5 times 10 to the 4 is 5 times 10000, or 50000.

If the number raised to a power is not 10, then the best you can do is approximate to a power of 10. For example, let's have a look at 2 to the power 100. By the rules of exponents, this is the quantity 2 to the power of 10, raised to the power of 10. Now 2 to the power of 10 equals 1024, which is roughly 10 to the power of three, so that 2 to the power of 10, raised to the power of 10 is about 10 to the power of thirty.

In general, logarithms to base 10 gtell you how many digits a number has. If the log to base 10 of a number is between n-1 and n, then the number has n digits. (Try this with some numbers between 1 and 10000.) The rules of logarithms can help. Going back to 2 to the power of 100, the number of digits is the first integer following the log to base 10 of the quantity 2 to the 100. This equals 100 times the log to base 10 of 2, which is 100 times 0.3010299957, or about 30.103, so 2 to the power of 100 is a number with 31 digits. Compare this with the rough approximation in the second paragraph.



Hi Kira.

What do you mean by "calculate" a number? Do you mean express it without exponents? Can you send me an example with a solution?

Yours is fairly straightforward because the exponent is on a 10 and we use base-10 numbering:

2 × 1035 is 2 with 35 zeroes after it.


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