If \(m + \cfrac{1}{m} = \sqrt{3}\), then find the simplest value of: (a) \(m^2 + \cfrac{1}{m^2}\), and (b) \(m^3 + \cfrac{1}{m^3}\).

Q.If \(m + \cfrac{1}{m} = \sqrt{3}\), then find the simplest value of: (a) \(m^2 + \cfrac{1}{m^2}\), and (b) \(m^3 + \cfrac{1}{m^3}\).

(a) \(m^2 + \cfrac{1}{m^2}\)
\(= \left(m + \cfrac{1}{m}\right)^2 - 2 \cdot m \cdot \cfrac{1}{m}\)
\(= (\sqrt{3})^2 - 2\)
\(= 3 - 2\)
\(= 1\) (Answer)

(b) \(m^3 + \cfrac{1}{m^3}\)
\(= \left(m + \cfrac{1}{m}\right)^3 - 3 \cdot m \cdot \cfrac{1}{m} \cdot \left(m + \cfrac{1}{m}\right)\)
\(= (\sqrt{3})^3 - 3 \cdot \sqrt{3}\)
\(= 3\sqrt{3} - 3\sqrt{3}\)
\(= 0\) (Answer)