It is proved that every pair of sufficiently large even integers can be represented in the form of a pair of equations, each containing two squares of primes, two cubes of primes, two biquadrates of primes, and $30$ powers of 2. Moreover, we also proved that every sufficiently large even integer can be expressed as the sum of two squares of primes, two cubes of primes, two biquadrates of primes, and $14$ powers of 2. These two theorems constitute improvements upon the previous results.