Apply logic part 4

${ 2 }^{ 2005 }\times { 5 }^{ 2000 }={ 2 }^{ 5 }.{ 2 }^{ 2000 }.{ 5 }^{ 2000 }\\ \quad \quad \quad \quad \quad \quad \quad \quad ={ 2 }^{ 5 }{ \left( 2.5 \right) }^{ 2000 }\\ \quad \quad \quad \quad \quad \quad \quad \quad =32.{ 10 }^{ 2000 }\quad \quad \\ now,\quad \\ { 10 }^{ 2000 }\quad contains\quad 2000\quad zeroes\quad after\quad 1\\ and,\quad \quad 32.{ 10 }^{ 2000 }=32\quad followed\quad by\quad 2000\quad zeroes.\\ \therefore \quad Total\quad no.\quad of\quad digits=(2000+2)=2002$

$\large{{ 2 }^{ 2005 }\times { 5 }^{ 2000 }\\ ={ { 2 }^{ 5 }\times 2 }^{ 2000 }\times { 5 }^{ 2000 }\\ ={ 2 }^{ 5 }\times { 10 }^{ 2000 }\\ =32\times \underbrace { 100000\dots 0 }_{ 2000\quad zeroes } \\ =\underbrace { 3200000\dots 0 }_{ 2000\quad zeroes } }$

Therefore the total number of digits is $\large \color{#20A900}{\boxed{2002}}$ .

Rewrite the given expression: $2^{2000} \times 2^5 \times 5^{2000}$ = $10^{2000} \times 32$ . The number $10^{2000}$ has 2000 zeros, and $1 \times 32$ has 2 digits. In total there are 2000 + 2 = 2002 digits.

$2^{2005} \times 5^{2000} = 2^{2000} \times 5^{2000} \times 2^{5} = 10^{2000} \times 2^{5} = 2000+2=2002$

2^5 × 2^2000 × 5^2000 = 32 × 10^2000 = 3.2 × 10^2001, so 2002 digits.

${ 2 }^{ 2005 }{ 5 }^{ 2000 }=\quad 2^{ 2000+5 }5^{ 2000 }$

$=\quad { 2 }^{ 5 }(2^{ 2000 }5^{ 2000 })$

$=\quad { 2 }^{ 5 }(2*5)^{ 2000 }=\quad { 2 }^{ 5 }(10)^{ 2000 }=32*{10}^{2000}$

We know that ${10}^{2000}$ contains 2000 zeros + one 1. Therefore it contains 2001 digits. However, when 32 is multiplied to ${10}^{2000}$ we get 32 followed by 2000 zeroes; so 2+2000 digits. Therefore, the number contains 2002 digits.

2^2005 x 5^2000 = 2^5 x 2^2000 x 5^2000 = 32 x (2 x 5)^2000 = 32 x 10^2000 Therefore, the answer has 2 + 2000 = 2002 digits.

((2^2000)x(5^2000))x2^5 =10^2000x32 so, we have total (2000+2) digits in total. here,2 is for 32&1

2^1 * 5^1 = 10 then 2 digits

2^3 * 5^3 = 1000 then 4 digits

so, num of digits = power + 1

2^2000 * 5^2000 then 2001 digits => 2000 zeros & 1 one

and , 2^5 = 32

when 32 * 1 , num of digits increase by 1

so the result is 2002 digits

2^2005 5^2000 =2^5 (2 5)^2000 =32 10^2000 since 10^2000 has 2000 zeroes followed by one 10^2000*32 has2002 digits