18. \( (2 p+q)^{3}-(p-q)^{3} \)

To simplify the expression \( (2p + q)^{3} - (p - q)^{3} \), we can apply the difference of cubes formula, which states that \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). Here, let \( a = 2p + q \) and \( b = p - q \). First, calculate \( a - b \): \[ a - b = (2p + q) - (p - q) = 2p + q - p + q = p + 2q \] Next, we need \( a^2 + ab + b^2 \). We can calculate each term: 1. \( a^2 = (2p + q)^2 = 4p^2 + 4pq + q^2 \) 2. \( b^2 = (p - q)^2 = p^2 - 2pq + q^2 \) 3. \( ab = (2p + q)(p - q) = 2p^2 - 2pq + pq - q^2 = 2p^2 - pq - q^2 \) Now combine them: \[ a^2 + ab + b^2 = (4p^2 + 4pq + q^2) + (2p^2 - pq - q^2) + (p^2 - 2pq + q^2) \] Combining like terms gives: \[ (4p^2 + 2p^2 + p^2) + (4pq - pq - 2pq) + (q^2 - q^2 + q^2) = 7p^2 + pq + q^2 \] Putting it all together into the difference of cubes formula: \[ (2p + q)^{3} - (p - q)^{3} = (p + 2q)(7p^2 + pq + q^2) \] And that's the simplified form!