Powers of Algebraic Fractions

How to compute the power of a fraction

The n-th power of an algebraic fraction is obtained by raising both the numerator and the denominator to the n-th power: $$ \left(\frac{A}{B}\right)^2 = \frac{A^2}{B^2} $$

A worked example

Let’s compute the square of the following fraction:

$$ \left( \frac{4y^3}{3x^2} \right)^2 $$

Since this is an algebraic fraction, its numerator and denominator are either polynomials or monomials.

To find its square, we raise both numerator and denominator to the second power:

$$ \frac{(4y^3)^2}{(3x^2)^2} $$

Now apply the laws of exponents:

$$ \frac{4^2 \, y^{3 \cdot 2}}{3^2 \, x^{2 \cdot 2}} $$

Simplifying gives the final result, which is the square of the original fraction:

$$ \frac{16y^6}{9x^4} $$

And so on.