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) \( m^{3}+\frac{1}{m^{3}} \)

To tackle these mathematical expressions, let's break them down step by step. a) Simplification: i) To simplify \( \left(m+\frac{1}{m}\right)\left(m^{2}-1+\frac{1}{m^{2}}\right) \): - We recognize that \( m^2 - 1 + \frac{1}{m^2} = (m^2 + \frac{1}{m^2}) - 1 \). - From the identity \( (m+\frac{1}{m})^2 = m^2 + 2 + \frac{1}{m^2} \), we find \( m^2+\frac{1}{m^2} = (m+\frac{1}{m})^2 - 2 \). - Thus, we expand \( (m+\frac{1}{m}) \left((m^2 + \frac{1}{m^2}) - 1\right) \). - Finally, substituting and simplifying leads to a clean expression. ii) To simplify \( \left(m+\frac{1}{m}\right)^{2} \): - Using the same identity as before, we get \( (m+\frac{1}{m})^{2} = m^2 + 2 + \frac{1}{m^2} \). b) Given \( m+\frac{1}{m}=2 \): i) To find \( m^{2}+\frac{1}{m^{2}} \): - From \( (m+\frac{1}{m})^2 = 2^2 = 4 \), we have \( m^2 + 2 + \frac{1}{m^2} = 4 \). - Therefore, \( m^2 + \frac{1}{m^2} = 4 - 2 = 2 \). ii) To find \( m^{3}+\frac{1}{m^{3}} \): - Use the identity \( m^3 + \frac{1}{m^3} = (m+\frac{1}{m})(m^2+\frac{1}{m^2}) - (m+\frac{1}{m}) \). - Substituting our known values, we get \( 2(2) - 2 = 4 - 2 = 2 \). Enjoy simplifying and calculating! 🧠✨