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It is given that ${{2}^{n-7}}\times {{5}^{n-4}}=1250$ , now we have to calculate the prime factorization of 1250.

"Prime Factorization" is finding which prime numbers multiply together to make the original number.

Now we have the number 1250.

So, $1250=2\times 5\times 5\times 5\times 5$

It can also be written as, $1250={{2}^{1}}\times {{5}^{4}}$.

Putting the value of 1250 in the form of prime factorization, we get

$\therefore {{2}^{n-7}}\times {{5}^{n-4}}={{2}^{1}}\times {{5}^{4}}$

Comparing the powers of 2 and 5 both sides, we get two equations

$\Rightarrow n-7=1\And n-4=4$

Calculating the values of $n$ from both equations, we get

$\Rightarrow n-7=1\Rightarrow n=7+1=8$

From first equation we get the value of $n=8$.

$\Rightarrow n-4=4\Rightarrow n=4+4=8$

Also, from the second we get the value of $n=8$.

Hence, as we are getting an equal value of $n$ from both the equation, which is equal to 8.