Francisco and Ryan are stuck simplifying radical expressions. Francisco has simplified to the quantity of x to the one half power, over x to the three eighteenth power. Ryan has simplified to the twenty seventh root of the quantity of x to the second times x to the third times x to the fourth. Using full sentences, describe how to fully simplify Francisco and Ryan's expressions. Describe if Francisco and Ryan started with equivalent expressions or if they started with expressions that are not equal.

X to the one half power, over x to the three eighteenth power is equal to x to the one half power, divided by x to the one sixth power, which equals x to the power of (one half minus one sixth), or x to the one third power.

The twenty seventh root of the quantity of x to the second times x to the third times x to the fourth equals the twenty seventh root of x to the ninth power which equals x to the one third power.

I cannot say whether Francisco and Ryan started with equivalent expressions but on final simplification they ended up the same. One might reasonably assume they started with equivalent expressions, but who knows if they made any mistakes in their simplifications.