Find the number of digits in the number

Question

2¹⁹⁹⁹ × 5¹⁹⁸⁹

Answer 1

If you know about logarithms, you know that you can express each of these numbers as a power of 10, using logarithms.  Then you can multiply them together and the result would be another number that's a power of 10, and that would be 95% of the way to the answer.

 

HOWEVER...  You can do this one without logarithms, just by knowing how to use the rules about exponents.

 

First notice that the exponent of 2 is just a little more than the exponent of 5.  You can split the term with 2 using the rule for multiplication of powers:

 

2¹⁹⁹⁹ = 2¹⁰ * 2¹⁹⁸⁹ 

 

Now go back to the original expression:

 

2¹⁹⁹⁹ × 5¹⁹⁸⁹ = 2¹⁰ * 2¹⁹⁸⁹ * 5¹⁹⁸⁹

 

Remember the rule about the product of a power -- it equals the power of the product.  So...

 

2¹⁰ * 2¹⁹⁸⁹ * 5¹⁹⁸⁹ = 2¹⁰ * (2¹⁹⁸⁹ * 5¹⁹⁸⁹) = 2¹⁰ * (2 * 5)¹⁹⁸⁹ = 2¹⁰ * 10¹⁹⁸⁹ 

 

2¹⁰ is 1024 -- you can see that on your calculator.  That's just a tiny bit more than 1000, which is 10³.

 

So 2¹⁰ * 10¹⁹⁸⁹ is just a bit more than 10³ * 10¹⁹⁸⁹, which is 10¹⁹⁹², again by the rule for multiplication of powers.  That's a 1 with 1992 zeroes after it, which is a number with 1993 digits.

 

So the answer is 1993.