In each of the following, perform the indicated operations...

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative.
$$\frac{\sqrt{48 x^{5} y^{4}}}{4 x^{2} y^{2}}-\sqrt{12 x}$$

Key Concepts

Domain Considerations

When simplifying expressions involving variables under radical signs, domain restrictions are essential to consider. Since the square root function is defined only for non-negative numbers, it is important to assume or verify that the variables inside the radical satisfy this condition. This ensures that the simplification process, such as removing absolute values or applying certain exponent rules, remains valid.

Rational Expression Simplification

Simplifying rational expressions entails reducing fractions by canceling out common factors in the numerator and the denominator. In the context of radicals, this may involve first simplifying the radical part independently and then reducing the resulting expression, ensuring that the expression is expressed in its simplest, most concise form.

Radical Simplification

This concept involves rewriting expressions containing square roots (or other roots) into their simplest form by factoring out perfect squares (or corresponding perfect powers in the case of other indices) from under the radical sign. The aim is to express the radical in terms of a product of an integer (or monomial with variables) and a square root of a simplified radicand, making further manipulation or comparison easier.

Exponent Rules

Understanding exponent rules is crucial when working with expressions under radical signs, particularly when variables are involved. This concept covers laws such as multiplying powers, dividing powers, and converting between radical notation and fractional exponents. These rules enable the extraction of variables raised to even powers from beneath the radical, thereby simplifying the expression.

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