I chose to use a fractional exponent because it is a powerful mathematical tool that allows us to represent roots and rational powers in a concise and efficient manner and show numbers that cant be represent in integer form such as irrational and undefeined numbers. Fractional exponents are a way of expressing powers and roots that are not whole numbers, essentially serving as shorthand form for roots and powers of numbers.

firstly fractional exponents are a way of using normal powers to represent the radical for example when we have an expression of the form ​, where “$a”$ is a base, $m$ is the numerator or the top number, and $n$ is the denominator or the bottom numbers, this represents the $n$th root of “$a”$ raised to the power of “$m”\; which\; is\; how\; many\; times\; you\; multiply\; itself\; onto\; itself\; with\; an\; index\; or\; root\; of\; “n”\; which\; is\; how\; many\; of\; the\; same\; number\; multipylied\; by\; each\; other\; is\; needed\; to\; equal\; this\; one$.

fractional radical also let you apply the laws of exponents, allowing for simplification of expressions and equations involving roots and powers allowing you to reduce the amount of steps required to use radicals in equation. for example we can multiply two numbers together and instead of converting them to the same form if one is an integer and the other is in fractional exponent form we then simply combine there powers together as shown in the below picture of product of powers and the fractional powers law an example would be instead of “” which takes other steps and is harder to solve we can translate it to this and use the product law which means we add the exponents when the bases are the same in multuplication “” which we know equals to or 64

furthermore, fractional exponents can also be shown as a negative powers allowing for more equation we can use radicals in and they can also be converted to radical form if required, further helping in simplification for example is equal to . Additionaly fractional exponents can represent irrational roots, providing a important tool in mathematical problem-solving to use them in a wide variety of way. For instance, ​ represents the cube root of 9, which is approximately 2.08, an irrational number. In summary, fractional exponents offer a simple and efficient way for expressing roots and rational powers of numbers, with a advandced understanding of their values significantly simplifying mathematical expressions and equations.

Exponent laws

photos: https://www.easysevens.com/laws-of-expeonents-and-logarithms/