dividing two rational expressions - What is the product rule and the quotient rule of dividing two rational expressions?

Sangeetha Pulapaka (last edited 4 years ago) (Expert, Educator @Qalaxia)

Radical expressions are written in simplest terms when

The index is as small as possible.
The radicand contains no factor (other than 1) which is the nth or greater power of an integer or polynomial.
The radicand contains no fractions.
No radicals appear in the denominator.

To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance,

If a and b are non negative, the square root of the product ab is equal to the product of the square roots of a and b

\sqrt{ab}= \sqrt{a}\sqrt{b}

This is the product rule of simplifying square roots

Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately.

The square root of quotient \frac{a}{b}= is equal to the quotient of the square roots of a and b where b≠0

\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}

This is the quotient rule of simplifying square roots

https://math.libretexts.org

Skills to recall :

What is a radical

https://www.themathpage.com/alg/radicals.htm

What is a radicand

https://www.themathpage.com/alg/radicals.htm

What is a denominator

https://www.mathsisfun.com/definitions/numerator.html

What is a numerator