Question

Calculate number of digits in 3^200 + 2^50

Answer 1

The number of digits in $3^{200}+2^{50}$ is 96.

Step-by-step explanation:

We have to find the number of digits in the expression $3^{200}+2^{50}$.

Taking each term individually;

The value of $3^{200}$ equals to $2.65614 \times 10^{95}$, this means that there are 96 digits in the value of $3^{200}$ because the power of 10 is 95 and there is also one digit before the decimal point which makes the total digits 96.

Or in other words, the value of log 3 to the base 10 is 0.477121, so multiplying this to 200; we get;

$0.477121 \times 200$ = 95.42 ≈ 96 digits.

Similarly, the  value of $2^{50}$ equals to $1.125899 \times 10^{15}$, this means that there are 16 digits in the value of $2^{50}$ because the power of 10 is 15 and there is also one digit before the decimal point which makes the total digits 16.

Or in other words, the value of log 2 to the base 10 is 0.3010299, so multiplying this to 50; we get;

$0.3010299 \times 50$ = 15.0515 ≈ 16 digits.

This means that adding 96 digits number to the 16 digits number, we get the 96 digits number only. So, the number of digits in $3^{200}+2^{50}$ is 96.

Answer 2

Given  : 3^200 + 2^50

To find : Number of Digits

Solution:

   N = 3²⁰⁰ + 2⁵⁰

=> N = (3⁴)⁵⁰  + 2⁵⁰

=> N =  81⁵⁰ + 2⁵⁰

83⁵⁰ = (81 + 2)⁵⁰  >  81⁵⁰ + 2⁵⁰

=>    81⁵⁰ < N  <  83⁵⁰

=>   a < N < b

a =  81⁵⁰

b = 83⁵⁰  

a = 81⁵⁰

=> log a = 50 log (81)

=> log a = 95.424

    10⁹⁵  <  a  < 10⁹⁶

=>  10⁹⁵  <  a

 

b = 83⁵⁰

=> log b = 50 log (83)

=> log b = 95.954

    10⁹⁵  <  b  < 10⁹⁶

=>   b  < 10⁹⁶  

 a < N < b  

=> 10⁹⁵  <  a  < N < b   < 10⁹⁶  

=> 10⁹⁵  <    N   < 10⁹⁶

Hence N has 96 digits

3²⁰⁰ + 2⁵⁰   has 96 digits

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