Question

How many digits are there in the number $2^3\times 5^4\times {20}^5$?

Answer 1

Consider, $2^3\times 5^4\times {20}^5$

$=2^3\times 5^4\times (2^2\times 5{)}^5$

$=2^3\times 5^4\times (2^2{)}^5\times 5^5$ ..... [Using product rule of exponents]

$=2^3\times 5^4\times 2^{10}\times 5^5$

$=2^{13}\times 5^9$

To find the number of digits in the given number, we can use log(base $10$)

$\log (2^{13}\times 5^9)=\log \left(2^{13}\right)+\log \left(5^9\right)$

$=13\log 2+9\log 5$

$=13\times 0.3010+9\times 0.6989$

$=3.9133+6.2907$

$=10.20$

Thus, there are $11$ digits