SOME HARD TRUTHS ABOUT FRACTIONAL EXPONENTS YOU NEED TO KNOW! PART 1

[@citytutoring]

ERRATA: In the last problem, there is a computational error on my part as I did not completely write out the indices. Thus: The expressions needs to be written as: 2^(13/7) and the second term can be rewritten as 2^(8/7), thus: 2^(13/7) * 2^(8/7) = 2^(13/7 + 8/7) = 2^(21/7) which equals 2^3 = 8.

[@bntns]

Great video. You explained the concepts well. Why though can't the denominator of the fractional exponent be (1 < x

[@citytutoring]

Thank you for the thoughtful question! Let's break it down: Understanding Fractional Exponents: Fractional exponents 𝑎^𝑚/𝑛 or (a^1/n)^m, where 𝑚 is the numerator and n is the denominator of the fraction. The key is that 𝑛, the denominator, must represent the "degree" of the root in its radical form. Restricting 𝑛: For 𝑎^𝑚/𝑛 to be expressed as a radical, n must be an integer because roots are defined by integer degrees (e.g., square root, cube root, etc.). If 𝑛 is not an integer (e.g., 1.5), the concept of a radical does not directly apply. In your example, if 𝑚 = 3 and 𝑛 = 1.5, simplifying m/n gives you 2. While a^2 is valid, it is not expressed as a radical/surd; it's just an integer exponent. Fractional 𝑛 as an Issue: If 𝑛 is not an integer, the concept of "root" as typically understood in radical notation breaks down. For instance, what does the "1.5-th root" of a number mean in radical terms? It doesn't map cleanly to a mathematical operation in the radical system. In the context of fractional exponents and radicals, the denominator 𝑛 needs to be an integer to properly represent a root. Cases 𝑛 ≤ 2: For 𝑛 < 2, fractional exponents like 𝑎^2/3 or 𝑎^1/2 are perfectly valid and correspond to cube roots and square roots. They can still be represented as radicals. The limit isn't at 𝑛 = 2. Rather, the issue arises when 𝑛 is not an integer. Radicals can only represent roots with integer indices. What About 2/5? The exponent 𝑎^2/5 is valid. It represents the fifth root of 𝑎^2 , or equivalently, (a^1/5)^2. The denominator 𝑛 = 5 is an integer, so it works perfectly in radical form. The key distinction is that 𝑛, not that it has to meet a specific lower or upper limit. I hope that settles the matter.

[@168grhp]

Most helpful question and explanation!

[@vidaroldfeldt4528]

[2^(13/7)]*[2^(8/7)] = 2^(21/7) = 2^3 = 8 ≠ 128

[@citytutoring]

Ah yes, thank you for bringing this to my attention, I realized I put in the other index. I will put a clarification in the description!

[@Nacho_Meter_Stick]

What about defining the exponent rules for the complex numbers?

[@citytutoring]

I plan to discuss the complex numbers and their exponents in another video series.

[@anacondaadultlearningcente4522]

Excellent lesson.

[@citytutoring]

Thank you!

[@RegiJatekokMagazin]

Im interested in theorical physics over my childhood. Im actually starting to understand how information bending in time space and entropy works. The basic principle of defining existence is, when we can define error, or what we "cannot do", in the language of nature.

[@petihundeuhun8950]

Magyarok itt is összefutnak. Kicsi a világ.😂

[@RegiJatekokMagazin]

@@petihundeuhun8950 Igen :)