Raising Rational Fraction to the Integer Power

In order to raise rational fraction `P/Q` to the natural `n`-power, we should raise to this power the numerator and denominator of fraction separately; the first expression is numerator and second expression is denominator of the result: `color(blue)((P/Q)^n=P^n/Q^n)` .

Example 1. Convert power into fraction: `((2x^2y^3)/(3z^5))^3`.

`((2x^2y^3)/(3z^5))^3=(2x^2y^3)^3/(3z^5)^3=(8x^6y^9)/(27z^15)` . When we raise fraction to the integer negative power we use the identity `(P/Q)^(-n)=(Q/P)^n`, that is correct for all values of variables, for which `P!=0` and `Q!=0`.

Example 2. Convert power into fraction: `(((a+b)^2(a-b)^3)/(a+2b)^4)^(-5)`.

`(((a+b)^2(a-b)^3)/(a+2b)^4)^(-5)=(((a+2b)^4)/((a+b)^2(a-b)^3))^5=(a+2b)^20/((a+b)^10(a-b)^15)` .