Exponents and fraction worksheets

I'm preparing for Discrete Mathematics in the Fall and would love feedback on this concept and resource list. I am familiar with data structures and algorithms and generally find math intuitive. After reviewing posts on our subreddit, here’s what I’ve compiled:
Key Algebra Concepts to Review:

• Exponents
• Fractions
• Polynomials
• Factoring and expanding
• Factorials
• Basic probability
• Rational vs. irrational numbers

Suggested Resources:

• Read: "Book of Proof" by Richard Hammack
• Watch: Trefor Bazett's Discrete Mathematics video series

Is this a good foundation? Let me know your thoughts.

I m 23, Currently i m learning Algebra 1 and i ve got stuck with these radicals and idk if should i move on, i understand them but it takes a while untill i m computing the operations and sometimes i m wrong.
Also yeah i don t know when should i move on, when to recap ? For 2 days i m practiging how to make the operations but i m still stuck at them and i don t want to loose my entire time on this neither to not knowing it. ( i can t. afford a tutor either)
Also i m learning in order to understand physics but everytime i m getting stuck like this i feel demoralised as it s such a long way, any advices , and how exactly did you cope with this? Anybody that was in the same situation like me? I need some encouragment so i can move forward or at least maybe some reality to step back
The thing is that if i m taking them individually i can solve them, but once they re adding more and more in the problem, (like fractions, exponents or just an unusual order than the standard one) or if simply change from the subject to another problem is like i forgot everything.
Also i m sorry for my english is not my primary language.

I have 2 worksheets in one workbook that are almost identical, but one has a few columns and tables hidden which in no way should be related to my issues. For context, the workbook is an insurance application, Sheet A to be filled by the applicant, Sheet B is where me and my team put in rates and other info we need to quote. So, I want all the values input on Sheet A to be duplicated in Sheet B with their corresponding cells.
Issue #1: I think I know why, but hoping there’s a work around .. The formula used to duplicate values from Sheet A to Sheet B do not translate with formulas in Sheet B. Ex.: Formula in cell on Sheet B to copy cell on Sheet A: ='Agent Filled SOV - Ex. Accts'!R5 & "" Formula on Sheet B using the value in the cell with the “copy” formula that now reads #VALUE and does not calc: =IF(R5,T5/R5,0)
Issue #2: Same scenario as Issue #1, but when I use the exact same “copy” formula in the 4 columns with Currency values listed, and the one column set as a Fraction, it freaks out! None of the Currency cells will read as currency, no matter what Number Format I choose for those cells/columns. Same goes with the decimal place for randomly some of the cells (even deep diving into the “More Number Formats …” section changes nothing), and only a few cells set as fractions actually are fractions and not decimals. The screens shots really explain this more.
Screenshots:
https://ibb.co/TgvrKJy https://ibb.co/ypm5JYP https://ibb.co/XtVzJj9 https://ibb.co/FsYRtcS
I really appreciate your help!!

Terms canceling out in fractions has always been a point of confusion for me. Mainly when you start mixing addition with multiplication, and especially when yous start involving variables and exponents.
For example, how would you simplify
6x+3 _______ x² +3
I honestly don't even know where to start with canceling out when addition gets involved, unless it is literal equations where I don't seem to struggle.
I also would like to ask for some problems that can be simplified for me to solve.
(Edited bc html hypertext)

Hi, I'm hoping to go back to school in the fall with college math classes. So I'm going through algebra again with the help of Professor Leonard (absolutely amazing) and other online resources.
I'm really trying to understand the math/formulas I'm learning, rather than just memorizing the formulas which is what I did a decade ago learning math.
So, in standard form m = -a/b. Rather than just memorize that, I want to understand why which is where I'm struggling.
If I get two points from the standard form equation, and use m = y2 - y1/x2 -x1 I get the same slope as m =-a/b, so I know it has to be true that it -a/b works. The y2 - y1/x2 - x1 version I get. Rise/Run which is based on the difference between the two points' X and Y axis on the line.
So why is it that in standard form -a/b works? It feels like that's saying opposite run/rise? I realize in this form we are using the coefficients of x and y rather than a specific point's x and y values like in the other format so it must have to do with that.
Bonus question if you have time -- I was playing around with a simple equation from a worksheet: 2x + 4y = 5. I found that after getting the slope and a point from that equation, and putting that in point-slope format, and then going back to standard form, I got 1/2x + y = 5/4 as my standard form equation. Which equates to the same line as the original problem. I can then multiply both sides by 4 to make it be the exact same as the original equation. So my question is are both of those equations of the line equally "correct"? They are the exact same line. Couldn't we then have an infinite number of line equations as long as we were doing the same thing to both sides of the equation? The writer of the original equation I'm guessing just got rid of the fractions to make it an easier question?
Thank you very much for any insight you can provide!!

Hi! I’m working on the Inequalities and Absolute Values chapter in the 5lb Book, but I don’t understand how a fraction with exponents was simplified in the answer key for that question and would love some help understanding.
I included a photo of the question and answer explanation, where you can also see how I was originally simplifying the fraction. Thank you!!

Hello everyone!
I am in adult high school math classes at 34 y/o and I just cannot accept math facts without understanding how they work or why they work. x^0=1, which I think I understand. What I would love is to see, is the same method to demonstrate this equation as being true, applied to numbers with greater exponents to kind of 'prove the rule'. I am an absolute noob, so feel free to explain it to me like I'm 5, in fact this is encouraged.
SO, demonstrating it as a fraction...
x^0 = x^0/x = 1
eg. 7^0/7 = 1
BUT
x^1 = x^1/1 = x
10^2 = 10^2/1 = 100/1 = 100
So, my problem is - we have to put X as the denominator for x^0 to equal 1. But to demonstrate the other exponents as fractions we must put 1 as the denominator of course. Why? Is it a standalone exception to any rule? Is there no rule that can be proved true across all exponents? What is going on. lol. help.

For each pair, which math type was more common on the last 10 tests?

• Geometric sequence or permutation?
• System of equations or absolute value?
• Shaded area or fractional exponents?
• Median or LCM?

I put together a list of the top 75 ACT questions, based on the research we do to maintain the Mathchops question base. This list is based on the last 10 tests and includes the most recent G19/Z13 TIR. Read on for the answers to the questions above.
Guaranteed To Show Up These have to be rock solid because A) they’ll definitely show up and B) they’ll often be combined with other skills.

1. Fractions and Decimals – All four operations. Mixed numbers.
2. Exponents – All operations. Fractional and negative exponents are very common too (see below).
3. Probability – Know the basic part:whole versions. There is usually a harder one also (like one with two events).
4. Negatives – Be comfortable with all operations.
5. Average – Also called the arithmetic mean. There is always a basic version and usually an advanced one, like the average sum trick (see below).
6. Linear Equations/Slope – Find the slope when given two points. Be able to isolate y (to create y = mx + b). All the standard stuff from 8th grade Algebra.
7. Quadratic skills – Factor. FOIL. Set parenthesis equal to zero. Graph parabolas.
8. Ratio – Part:part, part:whole.
9. Area/Perimeter of basic shapes – Triangles, rectangles, circles.
10. Percents – Know all basic variations. More advanced ones are common also.
11. Absolute Value – Sometimes basic arithmetic, sometimes an algebraic equation or inequality.
12. Picking Numbers – You never have to use this but it will be a useful option on every test.
13. Plug in answers – Like picking numbers, it’s not required but it’s often helpful.
14. Solving Equations – Be very comfortable with ax + b = cx + d. Distribute. Combine like terms. You also need to be able to create these equations based on word problems.
15. Radicals – Basic operations. Translate to fractional exponents.

Extremely Likely (> 80% chance)

1. FOIL – This has to be automatic.
2. SOHCAHTOA – Every variation of right triangle trig, including word problems.
3. Probability, two events – If there's a .4 probability of rain and a .6 probability of tacos, what is the probability of rain and tacos?
4. Average sum trick – 5 tests, average is 80. After the 6th test, the average is 82. What was the 6th test score?
5. MPH – The concept of speed in miles per hour is very common (sometimes combined with other conversion).
6. System of Equations – Elimination. Substitution. Word problems.
7. Composite function – As in g(f(x)).
8. Order of operations – Sometimes directly tested, other times part of a harder question.
9. Pythagorean Theorem – Sometimes asked directly, other times required as part of something else (like SOHCAHTOA or finding the distance between two points).
10. Time – Hours to minutes, minutes to seconds, time elapsed.
11. Angle chasing – 180 in a line. 180 in a triangle. Corresponding angles. Vertical angles.
12. LCM – Straight up. In word problems. In algebraic fractions.
13. Imaginary numbers – Powers of i. What is i^2? The complex plane.
14. Negative exponents – Know what they do and how to combine them with other exponents.

Very Likely (> 50% chance)

1. Factoring – Mostly the basics. Almost never involves a leading coefficient.
2. Fractional Exponents – Rewrite radicals as fractional exponents and vice versa.
3. Logarithms – Rewrite in exponential form. Basic operations.
4. Mixed Numbers – all four operations. Often combined with word problems.
5. Remainders – Can be simple or pattern based, as in “If 1/7 is written as a repeating decimal, what is the 400th digit to the right of the decimal point?”
6. Matrices – Adding, subtracting, multiplying. Knowing when products are possible.
7. Venn – There are 30 kids. 18 are in Algebra. 20 are in French. How many are in both?
8. Median – Middle when organized from low to high. Even number of numbers. What happens when you make the highest number higher or the lowest number lower?
9. Algebra LCD – Find the lowest common denominator, then combine the numerators.
10. Geometric sequence – You usually just need to find a subsequent term (not the formula).
11. Change the base – If 9^x = 27^5, what is x?
12. Given points, find equation – You’re given two ordered pairs and must find the linear equation.

Worth Knowing (> 25% chance)

1. Apply formula – they give you a formula (sometimes in the context of a word problem) and you have to plug stuff in.
2. Domain – Usually you can think of it as “possible x values”.
3. Given sine, find cosine – They give you one trig ratio and ask you to find another. As in, “If the sine of x is 4/5, what is the cosine?”
4. Law of Cosines – They almost always give you the formula. Then you just have to plug things in.
5. Midpoint – Given two ordered pairs, find the midpoint. Sometimes they’ll give you the midpoint and ask for one of the pairs.
6. Scientific notation – Go back and forth between standard and scientific notation. All four operations.
7. Shaded area – The classic one has a square with a circle inside.
8. Similar triangles – Relate the sides with a proportion.
9. Weird shape area – It’s an unusual shape but you can use rectangles and triangles to find the area.
10. Conjugates – Rationalize denominators that include radicals or imaginary numbers. Know that imaginary roots come in pairs.
11. Difference of two squares – (x + y)(x - y) = x^2 - y^2
12. Graph translations – Horizontal shifts, vertical shifts. Stretches. You should recognize y = 2(x+1)^2 - 5 right away and know exactly what to do.
13. Multistep conversion – For example, they might give you a mph and a cost/gallon and then ask for the total cost.
14. Parallelogram – Know that adjacent angles add to 180. Area formula.
15. Prime numbers – Usually combined with something else, like basic probability.
16. Probability with “not” – 3 reds, 5 blue, 6 green. Probability of picking one that’s not red?
17. Undefined – You can’t have 0 in the denominator.
18. Special right triangles – 30:60:90, 45:45:90.
19. Amplitude – If y = 5 sin(x) + 2, what is the amplitude?
20. Arithmetic sequence – Usually asks you to find a specific term, sometimes asks you to find the formula.
21. Expected value – There is a 0.3 chance of winning $100 in Game A and 0.2 chance of winning $200 in Game B, which is unrelated to Game A. If you place bets on both games, what is the expected value of your bets?
22. Volume of a prism – Know that the volume = area of something x height. Sometimes the base will be a weird shape.
23. Circle equations – (x-h)^2 + (y-k)^2 = r^2. Sometimes you have to complete the square.
24. Compare numbers – Be able to order square roots, decimals, and fractions.
25. Find inverse function – Switch y and x, then isolate y.
26. Permutation – You have 5 plants and 3 spots. How many ways can you arrange them?
27. Line of best fit – They’ll sometimes ask you to find the predicted value, or the difference between the predicted and actual values.
28. Linear inequality – Be comfortable solving algebraic inequalities. Graphs appear sometimes also.
29. Triangle opposite side rule – There is a relationship between an angle and the side across from that angle.
30. Inverse trig – Use right triangle ratios to find angles.
31. Toy Soldier (volume) – What happens to the height of the water when you drop an object in the bucket?
32. Use the radius – A circle will be combined with another shape and you have to use the radius to find the essential info about that other shape.
33. Periodic function graph – The basics of sine and cosine graphs (shifts, amplitude, period).
34. Value/frequency table – Find the median and mean in this format.

Basically the title. If I wanted the answer to the problem -4x^-5. Would it bes -20^-6 the answer or is it wrong?
If I wanted the answer for something like 1/2x² and 2x^1/2, how would I get it? Should I just do I normally with the fractions, so it be just x and x^-1/2. Since the exponent is negative, I remember that I should do something with it but I forgot, the only I thing I remember is that it should be 1/x²
Lastly the square root is confusing for me as well. How would I get the answer of /—x, /—3x, 3/—x, 3/–x² (they aren't negative, they're just the closest to a square root I could type) /- is the square root

Here's the vocab I have accumulated so far:
multiply = 곱하기
divide = 나누기
add = 더하기
subtract = 빼기
fraction = 분수
formula = 공식
What would the Korean vocab be for words like infinity, numerator, denominator, and exponent? I know there are way more math vocab but I'm struggling to think of them.
Edit: multiply = 곱하기 but it was written as 하기 which is clearly not right lol

tl;dr: As a rule of thumb the armour should be 1.5 to 2 times the mass of the ordnance, and the ΔV should be 3/5ths to 2/3rds the exhaust velocity* in order to best penetrate (laser based) point defense. Assumptions and derivation to follow:
\exhaust velocity and mass-specific impulse are different things really but I'll use them interchangeably*
​
The goal is to get a payload to a target with the lightest possible volley of missiles. Factors that affect this value are:

• Missile mass (m)
• Closing velocity - higher ΔV (change in velocity) means proportianally less time for point defense to shoot you down.
• Armour thickness (t) - at least against lasers, armour thickness is directly proportional to how long it'll last under fire.

Based on this we can construct a cost function G to be minimised (assuming zero inital velocity):
[1] G = m / (ΔV * t)
m is a function of payload mass (P), armour mass (A), the dry mass fraction of your propulsion system (near 0 for most user designs, which will make this simpler), mass-specific impulse (c) and ΔV:
[2] m ≈ (A + P) * exp(ΔV / c)
t is a function of armour mass and surface area. The way surface area scales with missile mass depends on wether the missile is scaled omnidirectionally (preserving shape) or radially (preserving thrust to mass ratio), we can account for this with a scaling exponent n which is ~2/3 and ~1/2 respectively.
[3] t ∝ A / m^n = A / [ ( A + P ) * exp( ΔV / c ) ]^n
Substituting into eq 1. results in:
[4] G = [1 / ( A * ΔV )] * [( A + P ) * exp( ΔV / c )]^(n + 1)
I'll spare you the differentation, but suffice it to say this function reaches a minima when:
[5] A = P / n
and:
[6] ΔV = c / (n + 1)
​
I like to design my missiles to have the same thrust to mass ratio because it means when I've tuned a controller for one I know it'll work with minimal tweaking for the others, so I take n = 1/2. Armour is polyethylene if I'm being cheap, aramid fiber otherwise, and ordnance is Iridium flak with the number of shrapnel pieces adjusted to reduce overpenetration.
Hope this is handy.

For emulator authors and other color conscious persons around here, these are some algorithmically created Apple II colors in sRGB that don't do the following things, unlike a lot of other sets of Apple II colors used in many places.
So here is what they DON'T do.

• confuse any of YUV, YIQ, YCbCr
• confuse non-linear with linear light
• confuse the Y of CIE 1931 XYZ values with the Y of YUV, YIQ or YCbCr.
• confuse different standard gamma formulas with each other or with a naive power function
• use the 1953 standard for NTSC, which was no longer used in practice in the late 1970s or 1980s (instead it uses the mid-1980s "SMPTE-170M" standard for NTSC)
• confuse sRGB with 1953 NTSC RGB, with the 1980s NTSC standard SMPTE-170M RGB, or with any other RGB color space
• use bad color primaries or white points, or ignore such issues altogether
• ignore the NTSC standard's 7.5 IRE "pedestal" of the black level above the sync level
• confuse any of the following signal shapes: sine waves, square waves, rectangular waves
• use relative values of U and V (or I and Q) when absolute values are perfectly available through Fourier Series expansion of the rectangular or square waves
• use any low-exactness constants. All constants used in the code are either integers, fractions of integers, or exact-exponent powers or roots of fractions of integers, thus the code can work at any accuracy
• use any old values for the NTSC monitor knobs "brightness", "picture", "color" and "hue". The values are carefully calculated to find a set of colors for 8-Bit Apples that is as close as possible to the one and only official, exact set of colors Apple ever published for the Apple II - those of the Apple IIGS published in IIGS Technical Note #63, and incidentally also used by the Software of the Apple //e card for the Mac LC.
• calculate color differences in a naive way, such as the root of squared differences of red, green and blue. Instead, it uses the CIEDE2000 formula, a pretty complex calculation based on the L*a*b* color space, which is currently the best standardized color difference formula available.

The resulting values can be found below. If interested, I can add the code to create them, written in GNU bc for convenient high-precision math; GNU bc is available on any decent Linux system and also in MSYS2 for Windows. Are you allowed to post code here?
You can see the colors in the attached Images; note that these will only look approximately right unless your Monitor is calibrated to sRGB, which is the closest equivalent to a common internet color standard that we have. On many un-calibrated monitors, the colors will look slightly too blueish.
Here are the results in text form. Left side is 8-bit Apple II, right side is Apple IIGS.
brightness= 0.04585129703904686
picture = 0.89208098132025144
color = 0.78486602944212231
hue =-0.64593628843128830
RMS ∆E =13.99837234804931772
*BASIC** **CIE 1931 xyY Apple //e** **sRGB* **sRGB* **CIE 1931 xyY Apple IIgs* *CIEDE2000
-------- -------------------------- ------- ------- -------------------------- ----------
COLOR= 0 x=0.3127 y=0.3290 Y=0.0000 #000000 #000000 x=0.3127 y=0.3290 Y=0.0000 ∆E= 0.0000
COLOR= 1 x=0.5045 y=0.2742 Y=0.0864 #9F1B48 #DB1F42 x=0.5693 y=0.3055 Y=0.1641 ∆E=14.7607
COLOR= 2 x=0.1746 y=0.0899 Y=0.0966 #4832EB #0C11A4 x=0.1549 y=0.0699 Y=0.0316 ∆E=13.0275
COLOR= 3 x=0.2802 y=0.1499 Y=0.2565 #D643FF #DC43E1 x=0.3142 y=0.1719 Y=0.2465 ∆E= 4.6280
COLOR= 4 x=0.2712 y=0.4639 Y=0.1340 #197544 #1C8231 x=0.2903 y=0.5283 Y=0.1630 ∆E= 7.2075
COLOR= 5 x=0.3126 y=0.3290 Y=0.2189 #818181 #636363 x=0.3127 y=0.3289 Y=0.1258 ∆E=11.6435
COLOR= 6 x=0.1902 y=0.1821 Y=0.2873 #3692FF #393DFF x=0.1667 y=0.0892 Y=0.1155 ∆E=27.1885
COLOR= 7 x=0.2604 y=0.2172 Y=0.4175 #B89EFF #7AB3FF x=0.2261 y=0.2335 Y=0.4374 ∆E=18.0221
COLOR= 8 x=0.3740 y=0.5439 Y=0.1080 #496500 #916400 x=0.4867 y=0.4541 Y=0.1504 ∆E=23.8370
COLOR= 9 x=0.5335 y=0.4168 Y=0.2680 #D87300 #FA7700 x=0.5512 y=0.4027 Y=0.3337 ∆E= 6.6844
COLOR=10 x=0.3126 y=0.3290 Y=0.2189 #818181 #B3B3B3 x=0.3127 y=0.3289 Y=0.4480 ∆E=15.8373
COLOR=11 x=0.3774 y=0.2822 Y=0.4359 #FB8FBC #FBA593 x=0.4113 y=0.3485 Y=0.4942 ∆E=19.0780
COLOR=12 x=0.3143 y=0.5915 Y=0.4381 #3CCC00 #40DE00 x=0.3133 y=0.5923 Y=0.5291 ∆E= 4.3511
COLOR=13 x=0.3980 y=0.5240 Y=0.5865 #BCD600 #FEFE00 x=0.4208 y=0.5066 Y=0.9134 ∆E=11.5941
COLOR=14 x=0.2666 y=0.3908 Y=0.6271 #6CE6B8 #67FCA3 x=0.2786 y=0.4457 Y=0.7511 ∆E= 9.2471
COLOR=15 x=0.3126 y=0.3290 Y=0.8788 #F1F1F1 #FFFFFF x=0.3127 y=0.3289 Y=0.9999 ∆E= 2.8580

While doing some wikipedia reading on fractional derivatives, i encountered the idea of finding a functional square root of a derivative, and this got me wondering about some properties and generalizations of the notion of the order of derivatives. First, since the notation of derivatives is analogous to that of exponents, are there deeper connections tying these two operators together. I know there is a connection based on the idea of repeating the same operation n number of times on a value/function to obtain an output value/function, but I was wondering if there were any deeper connections or shared properties between the order of a derivative and the power of an exponent.
Another thing i was curious about was if you could generalize the notion of non-integer orders of derivatives even further. Like we can make fractional derivatives through defining the order of a derivative on the rationals, but is there a way to take this further. Like would it make any sense to define the order on the reals, complex numbers, matrices, p-adics, higher dimension extensions of these other number system, or other fields/rings/ groups that are not direct extensions of the integers.

Hello! I am a newish teacher who is looking to revamp my AP chemistry curriculum here at my school because it has not been going well the past couple of years. We get through only a fraction of the required content for the AP test. I went to my admin to discuss getting a pre-made currriculum that I can edit and they said they will pay for one for me as long as it is not from Teachers Pay Teachers. This is unfortunate as I was eyeing some really good looking curricula on TPT but I unfortunately cannot fork over my own $300-$600 at this current moment. Are there any other resources that you all are aware of where I can purchase or find an entirely made curriculum (tests, notes, worksheets, etc). I know I am competent and able to make my own brand new curriculum from scratch but we are about to become foster parents and I want to save my time, energy, and peace for that. Any suggestions would be greatly appreciated!!

Hello all,
So, I just took my final for pre-calc and I can most certainly say I failed. With that, I want to self teach pre-calc over the summer.
I’m looking for a textbook, one that covers all of pre-calculus. I’m also looking for a textbook for “fundamental” math. Something I realized is that I freeze up when dealing with exponents, fractions, factoring, and other fundamental maths.
All recommendations are welcome. Paid or free textbooks are fine as well. Thanks.

I have two problems that I don't know how to solve y = 3x^5/4 + x and -4x^-5.
My best guesses are 15/4x^4/3 + 1 and 1/-4x⁵

We are a Client Accounting Services and Fractional CFO firm. I am looking for a system to help us with our internal close worksheets, workflow management across the team, and a Client Portal for document management etc. What systems do you like and why? Seeking your feedback and experiences with the systems, what did you like, not like, etc. Greatly appreciate your help, thank you in advance
View Poll

[Sorry if this is a repost, my previous attempt got "removed by Reddit's filters" so I'm trying again]
​
To start, I will make it clear that I don't know anything about imaginary/complex numbers/whatever 'i' is, and I am not learning about them (for now). Specifically, I'm currently just going over graphing functions.
So, if I put (-1)^(1/3) in a calculator, it tells me the answer is -1. This makes sense to me; ^(1/3) is the same as a cubic root, and (-1)*(-1)*(-1) = -1. This seems to be the case for 1/[any odd integer].
If instead I put (-1)^(2/3) in a calculator, it tells me the answer is undefined (or it uses 'i'). This kind of makes sense to me? The way I read it is that -1 is squared, which turns it to +1, and then the cubic root of that is +1. Or you find the cubic root of -1, and then square that, which also is +1. But Because you can't reverse this process, it doesn't count as a 'real' answer? I am aware that this probably isn't a logically sound way of going about it, it's just me trying to understand why it gives that answer.
So then why does (-1)^(3/5) not just equal to -1? the 5th root of -1 is -1, and -1 cubed is also -1. So shouldn't it just be -1, rather than a complex number (or just "error 2" according to my SHARP calculator)? Also, like I mentioned, I'm currently practicing graphing functions, and according to desmos, y = x^(3/5) IS defined for negative numbers, and it is equal to -1 at x = -1. So seemingly (-1)^(3/5) is actually defined. It also says that x^(2/3) is actually defined for negative numbers, they just always equal a positive number (so -1^(2/3) = positive 1). I think it might be okay to just use the rule-of-thumb that if the denominator is even, then it is undefined, but I'm not certain if that's actually correct.
I hope someone can help clear this up for me. Or, at the very least, just give me a clear and concise rule for evaluating negatives bases with fractional exponents, if such a rule even exists. As far as I can tell, my current rule-of-thumb is enough to get me by for now while I'm just doing graphs of functions, but will become an issue in the future when I move on to other topics. Thank you!

This week, I was able to learn about methods to represent data, which includes binary, octal, hexadecimal, and base-27. Although I already knew what binary was, I never learned how to perform different mathematical operations on them, nor did I know how to represent fractions, exponents, and negative numbers. I also didn't know base-8 and base-16 existed until I remembered seeing a Windows error line that started with "0x" (base-16 format).
Before this week, I associated binary as yes/no or on/off values, because in many flip-flop circuits, this is the case. I now know that binary is more than just a signal input/output, it is a representation of data. I was also looking into my previous research on a 4-bit binary counter (74LS93), and finally understood why it can only count from 0 to 15 (base-10).
1 bit = 1 binary digit
Largest Representable Unsigned Binary Number in 4-bits = 1 1 1 1
Smallest Representable Unsigned Binary Number in 4-bits = 0 0 0 0 = 0
Converting to Base-10...
1 x 2^0 + 1 x 2^1 + 1 x 2^2 + 1 x 2^3 = 1 + 2 + 4 + 8 = 15
I was also able to live-code in class. Though I didn't fully understand how binary conversion worked at the time, my classmates were able to walk me through the process, which helped me greatly and made the experience less daunting. Thank you to all of those who helped!