7 Given that \( m+n=9 \) and \( m n=8 \), find the value of \( \begin{array}{ll}\text { (a) } m^{2}+n^{2}, & \text { (b) }(m-n)^{2}\end{array} \)

We are given: - \( m+n=9 \) - \( mn=8 \) **Step 1: Calculate \( m^2+n^2 \)** Recall the identity: \[ (m+n)^2 = m^2 + n^2 + 2mn \] Substitute the known values: \[ 9^2 = m^2+n^2+2(8) \] This simplifies to: \[ 81 = m^2+n^2+16 \] Solve for \( m^2+n^2 \): \[ m^2+n^2 = 81-16=65 \] **Step 2: Calculate \( (m-n)^2 \)** Recall the identity: \[ (m-n)^2 = m^2+n^2-2mn \] We already found \( m^2+n^2=65 \) and we know \( mn=8 \). Substitute these values: \[ (m-n)^2 = 65-2(8) \] \[ (m-n)^2 = 65-16=49 \] **Final Answers:** \( m^2+n^2=65 \) \( (m-n)^2=49 \)