Radikalform

Radikalform, or radical form, is a way of representing numbers or expressions that involve roots so that the radicals are in their simplest, standard form. In radical form, a radical expression is written with the radical symbol (such as a square root or higher root) and any factors that can be pulled outside the radical have been extracted. The goal is to express each radical in terms of a radical with a square-free (for square roots) or otherwise simplified radicand.

For square roots, the radicand is simplified by factoring it into perfect square factors and non-square factors. For example, sqrt(72) = 6 sqrt(2) because 72 = 36 × 2, and sqrt(36) = 6. Similarly, sqrt(12) = 2 sqrt(3). The same idea extends to higher roots: sqrt[n]{a} can be written as r sqrt[n]{d} where a = r^n d and d is not divisible by any nth power greater than 1.

Radical form is also used to simplify expressions with several radicals and to rationalize denominators. For instance, 1/√5 can be rewritten in radical form without a radical in the denominator as √5/5. When combining terms, like x√2 + 3√8, recognizing that √8 = 2√2 allows simplification to (x + 6)√2.

In summary, radical form standardizes expressions involving roots by ensuring radicals are as simple as possible and outside factors are fully extracted, aiding comparison, computation, and further manipulation in algebra and calculus.