You should use the following exponential law, such that:

`a^(x+y) = a^x*a^y`

Reasoning by analogy yields:

`3^(1+log_3 7) = 3^1*3^(log_3 7)`

You also need to use the following exponential and logarithmic identites, such that:

`log_a b^n = n*log_a b`

`a^(b*c) = (a^b)^c`

Reasoning by analogy yields:

`log_4 121 = log_4 11^2 => log_4 11^2 = 2log_4 11`

Replacing `2log_4 11` for `log_4 121` yields:

`2^(log_4 121) = 2^(2log_4 11) = (2^2)^(log_4 11)`

You need to use the following logarithmic identity such that:

`a^(log_a b) = b`

Reasoning by analogy yields:

`3^(log_3 7) = 7`

`4^(log_4 11) = 11`

Performing the required transformations, you may evaluate the given number such that:

`3^(1+log_3 7) - 2^(log_4 121) = 3*7 - 11 = 21-11 = 10`

**Hence, evaluating the given number, using the indicated logarithmic and exponential identities, yields **`3^(1+log_3 7) - 2^(log_4 121) = 10.`