Convert radicals to and from fractional indices.

Simplify $\sqrt[5]{x^2}$ by converting to fractional index form.

A structural engineer uses the formula \( L = k \sqrt[3]{M^2} \cdot \sqrt{d} \) to determine the load-bearing capacity \( L \) (in kilonewtons) of a reinforced concrete beam, where \( M \) is the mass of steel reinforcement (in kilograms), \( d \) is the beam depth (in metres), and \( k \) is a material constant. (a) Express \( L \) in the form \( L = k M^a d^b \), where \( a \) and \( b \) are rational numbers. (2 marks) (b) A beam with \( M = 64 \) kg and \( d = 0.25 \) m has a load-bearing capacity of \( L = 80 \) kN. Determine the value of the material constant \( k \). (2 marks) (c) Hence, determine the mass \( M \) of steel reinforcement required for a beam of depth \( d = 0.64 \) m to achieve a load-bearing capacity of \( L = 128 \) kN, using the value of \( k \) found in part (b). (1 mark)

Simplify the following expressions, giving your answers in fractional index form. (a) $\sqrt[3]{x^5}$ (b) $\frac{2}{\sqrt{a^3}}$ (c) $\sqrt[4]{16m^8}$