5. Complete the following. a) Simplify: i) \( \left(m+\frac{1}{m}\right)\left(m^{2}-1+\frac{1}{m^{2}}\right) \) ii) \( \left(m+\frac{1}{m}\right)^{2} \). b) If \( m+\frac{1}{m}=2 \), calculate the value of: i) \( m^{2}+\frac{1}{m^{2}} \) ii) \( m^{3}+\frac{1}{m^{3}} \)

Let's solve each part of the problem step-by-step. --- ### **Problem 5** **a) Simplify:** **i) \( \left(m + \frac{1}{m}\right)\left(m^{2} - 1 + \frac{1}{m^{2}}\right) \)** To simplify the expression, we'll expand the product by multiplying each term in the first parenthesis with each term in the second parenthesis: \[ \begin{align*} \left(m + \frac{1}{m}\right)\left(m^{2} - 1 + \frac{1}{m^{2}}\right) &= m \cdot m^{2} + m \cdot (-1) + m \cdot \frac{1}{m^{2}} \\ &\quad + \frac{1}{m} \cdot m^{2} + \frac{1}{m} \cdot (-1) + \frac{1}{m} \cdot \frac{1}{m^{2}} \\ &= m^{3} - m + \frac{1}{m} + m - \frac{1}{m} + \frac{1}{m^{3}}. \end{align*} \] Now, combine like terms: \[ m^{3} - m + m + \frac{1}{m} - \frac{1}{m} + \frac{1}{m^{3}} = m^{3} + \frac{1}{m^{3}}. \] **Answer:** \[ \left(m + \frac{1}{m}\right)\left(m^{2} - 1 + \frac{1}{m^{2}}\right) = m^{3} + \frac{1}{m^{3}}. \] --- **ii) \( \left(m + \frac{1}{m}\right)^{2} \)** To simplify the square of a binomial, use the formula \((a + b)^2 = a^2 + 2ab + b^2\): \[ \begin{align*} \left(m + \frac{1}{m}\right)^{2} &= m^{2} + 2 \cdot m \cdot \frac{1}{m} + \frac{1}{m^{2}} \\ &= m^{2} + 2 + \frac{1}{m^{2}}. \end{align*} \] **Answer:** \[ \left(m + \frac{1}{m}\right)^{2} = m^{2} + 2 + \frac{1}{m^{2}}. \] --- **b) Given that \( m + \frac{1}{m} = 2 \), calculate the value of:** **i) \( m^{2} + \frac{1}{m^{2}} \)** Start with the square of \( m + \frac{1}{m} \): \[ \begin{align*} \left(m + \frac{1}{m}\right)^{2} &= m^{2} + 2 \cdot m \cdot \frac{1}{m} + \frac{1}{m^{2}} \\ &= m^{2} + 2 + \frac{1}{m^{2}}. \end{align*} \] Given that \( m + \frac{1}{m} = 2 \), square both sides: \[ 2^{2} = m^{2} + 2 + \frac{1}{m^{2}} \\ 4 = m^{2} + 2 + \frac{1}{m^{2}}. \] Subtract 2 from both sides to find \( m^{2} + \frac{1}{m^{2}} \): \[ 4 - 2 = m^{2} + \frac{1}{m^{2}} \\ 2 = m^{2} + \frac{1}{m^{2}}. \] **Answer:** \[ m^{2} + \frac{1}{m^{2}} = 2. \] --- **ii) \( m^{3} + \frac{1}{m^{3}} \)** To find \( m^{3} + \frac{1}{m^{3}} \), use the identity: \[ \left(m + \frac{1}{m}\right)^{3} = m^{3} + 3m + 3\cdot\frac{1}{m} + \frac{1}{m^{3}}. \] Rearrange to solve for \( m^{3} + \frac{1}{m^{3}} \): \[ m^{3} + \frac{1}{m^{3}} = \left(m + \frac{1}{m}\right)^{3} - 3\left(m + \frac{1}{m}\right). \] Given \( m + \frac{1}{m} = 2 \): \[ m^{3} + \frac{1}{m^{3}} = 2^{3} - 3 \cdot 2 = 8 - 6 = 2. \] **Alternatively**, since \( m + \frac{1}{m} = 2 \), solve for \( m \): Multiply both sides by \( m \): \[ m^{2} + 1 = 2m \\ m^{2} - 2m + 1 = 0 \\ (m - 1)^{2} = 0 \\ m = 1. \] Then, \[ m^{3} + \frac{1}{m^{3}} = 1^{3} + \frac{1}{1^{3}} = 1 + 1 = 2. \] **Answer:** \[ m^{3} + \frac{1}{m^{3}} = 2. \] --- ### **Final Summary** **a) Simplifications:** - **i)** \(\left(m + \dfrac{1}{m}\right)\left(m^{2} - 1 + \dfrac{1}{m^{2}}\right) = m^{3} + \dfrac{1}{m^{3}}\) - **ii)** \(\left(m + \dfrac{1}{m}\right)^{2} = m^{2} + 2 + \dfrac{1}{m^{2}}\) **b) Given \( m + \dfrac{1}{m} = 2 \):** - **i)** \(m^{2} + \dfrac{1}{m^{2}} = 2\) - **ii)** \(m^{3} + \dfrac{1}{m^{3}} = 2\)