VIDEO: Writing the root as a power

Writing roots as potencies - this is how it works

• Roots, whether the simple square root or higher roots, are not only unwieldy, but you can In many cases, you can only count on it under difficult conditions, which also quickly leads to errors sneak in.
• But: Every root can be converted into a power, with the corresponding exponent for roots being a fraction. For these powers, however, the relatively clear power laws apply, with which roots can also be treated and often even simplified (see examples below).
• The following applies: ⁿ√ a = a ^{1 / n} (read: the nth root of a is a to the power of 1 / n).
• You write accordingly for √3 = 3 ^1/2 respectively. 3 ^0,5  and for x ^1/6 = ⁶ √ x.
• Even more complicated root expressions can be written as powers in this way. For example (follow the power laws) ⁵ √ x³ = (x³)^1/5 = x ^3/5.


Derive 2 by x - this is how it works with fractional-rational functions

If you want to derive the function "2 by x", you can do this with a little ...

• The last example in particular makes it clear that the power notation for complex root expressions not only creates an overview and makes arithmetic easier, but that it is also based on the
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  calculator in this way complex roots simply and easily with the x^yLet the button be pulled. Depending on the model, you have to use a fraction or a fraction for y. enter a decimal number.
• Any why is this the case? Here, too, of course, mathematicians want to ensure that the calculation rules that apply to powers are retained. For example, according to the root definition (ⁿ√ a) ⁿ = a. According to the power laws we get 1 / n x n = 1. So the definition is consistent. Just by the way!

Calculating with "fractions" - examples

Many denote root as "fractional powers". Of course, this is not entirely true, even if roots turn out to be Potencies display with fractions as exponents. In the following, three examples are used to show how calculating with such "fractional powers" can easily be derived from the power laws:

• One computes √a³ _* √a = a³/ 2 * a¹/ 2 = a⁴/ 2 = a² (Add potencies when taking and then reduce the potency).
• So is ⁴√ a^-2 = a^-2/4 = a^-1/2 = 1 / √a (also use the definition of negative exponents).
• It is (ⁿ√ a²)ⁿ = (a²/n)ⁿ = a²n / n = a² (shorten in potency).