(- 4 cubed / (-4) squared ) to the negative 2 power=? How to do this! MANY will get WRONG!
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- Опубликовано: 8 сен 2024
- Simplify a fraction with powers and negative exponents with no calculator.
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Personally, I love the long-winded explanations. They really clarify the particular topic discussed, and my namesake John turns Math from something almost frightening to something friendly and welcoming. I'm 64, and Math has always been scary, particularly things like Algebra and Calculus, which I never studied as I left school to enter the workforce, but to my amazement and his credit, he made me begin to understand both within a couple of minutes of explanation. Well done, mate, regards from Christchurch NZ.
SPURIOUS LONGWINDED EXPLANATIONS ARE NOT AESTHETICand good mathematics is always transparent and therefore lucidly clear
I remembered a movie called Stand and Deliver about a Hispanic teacher who went and Ap calculus in a LA school I think his name was Jaime Escalante the one scene I remembered was him saying " A negative times Negative equals a positive" he had the class repete this three times. He took a class of failing kids in his first year and they all passed but had to take the test again because the powers thought they cheated because of the work through were almost identical, they retook the test and some got a better score the the first time crazy right this just shows that a good teacher is worth more then they are getting paid. R.IP Mr. Escalante he passed last year I think ( movie stared Edward James Olmos as Mr. Escalante and Lou Diamond Phillips in 1988)
I was just looking at this math, and didn’t know to do. I’m adult student doing the GED. Thank you for breaking down the maths for me I really appreciate you❤
Parabéns por estudar depois de adulto. É difícil mas não é impossível. Força e resiliência pra você ! 👏
Good Luck to you! I think that is exactly why he makes these videos.
Much of the problems in this task is not only about mathematical logic as such, but about conventions of notation. I got it "right", but I had to think somewhat about what parts of the notations meant.
BTW, the problem could have been more complicated if the powers weren't chosen such that sign ambiguity due to order of operations didn't mask it. For example, -4^3 could have two interpretations, either -(4^3) or (-4^3). But in this case, it doesn't matter. If a positive exponent were used, then the order of operations would matter. And then in this case, if the "outer" exponentiation would have been an odd negative power, e.g., -3, then this sign would matter.
Yeah, many will have gotten this right for the wrong reasons. Should have been an even number in the exponent. His claim that not knowing this would lead to the wrong answer makes no sense.
Yes, it does seem a missed opportunity. He might just as well have had simply 4³ / 4².
The two possible interpretations of -4³ aren't -(4³) and (-4³) though. Those are the same interpretation. The two possible interpretations are -(4³) and (-4)³.
1/16 , in my head in under 10 secs!! :) I am 85 y.o. and never went to Uni - but I have always adored mental maths and algebra!! ( And I say maths as a Brit!!! )
Yes, we are to old for unnecessary complications😂
4^3=4^2*4 ALSO
[-4^3/(-4)^2]= - 4
(-4)^-2= 1/4^2=1/16
i am 72
In my math courses, -4^3 was to be treated as (-4)^3, treating the minus as an unary operator, and unary operators have the highest priority, before parentheses. And, thus, -4^2 = -4 * -4 = 16, not -16.
The sign inside the bracket don't mater. The result can only be positive.
So... -4^3÷-4^2 = -4^(3-2) = -4^1 = -4.
Finally, as x^-1 = 1/x. We finish this way, 1/(-4^2) = 1/16
In a test with multiple answers, if you don't know an answer, go to the few next ones. I very often found that subsequent question provided hint for previous ones, how to solve a previous question, if not the actual answer.
I will get it wrong cuz I have NO IDEA what to do😅
Thanks for the refresher!
-4^-2=1/16
The exponent of the numerator is 2Z+1 so (-B)^E will be the same as -(B^E) both by value and signage, so you can't really make a mistake here.
You could make an error in the numerator if the parenthesises wasn't there, as -(B^2Z) is negative and (-B)^2Z is positive.
It is even worse if E is not an integer as (-B)^E would then be complex number.
But the signage of the fraction doesn't actually matter as the outer exponent is 2Z, and any real number to the 2Z will be a positive real number, which means that you can completely ignore any signage inside any of brackets.
The only thing that can go wrong here is if you don't know what X^(-Y) = 1/(X^Y).
With that knowledge you can rewrite the problem to 1/((4^3)/(4^2))^2 = 1/(4^(3-2))^2 = 1/(4^1)^2 = 1/(4^2) = 1/16.
Got it. A.... the final exponent makes it a fraction. Thanks for the fun.
correct, negative exponents tell you that they are "under the line"
It was a good well timed refreasher,thank you.😊
I got it right. I was a math teacher and a lady engineer.
i always get your problems right buy this is the first one to stump me. I guess I needed that little review of using negative powers
I effed up at the last step, forgetting -1/4 was in parentheses. I did -1/4 ^2 (which is -1/16) instead of (-1/4)^2.
Alternativa (a) 1/16
Boa questão !!! 🇧🇷
-4^-2 =-1/4^2 = 1/16 positivo.
You took an outrageous number of words to explain that with the neg 4 cubed, there are implied parentheses around the 4 cubed...such that it is -1 x the 4 cubed. In fact, dozens of your problems are constructed this way. In fact, there is no reason to write problems this way when the addition of parentheses, in the right places, & in most cases, eliminates the need to remember PEMDAS,
I got -4^-2 = -4^ =-16 =1/16 . Sometimes I don't like your Methods. I use the index law. ( Ex Maths teacher From Australia)
Where did you solve for -4 exp 3 / -4 exp 2, both numerator and denominator to exp -2? 3 x-2=-6; 2 x-2=-4. -4 brought up to the numerator = -2 (-6 + 4). So, -4exp -2 = 1/-4exp 2 = 1/16.
-4^3 = -64. -4^2 = 16. -64 / 16 = -4
Boy you really stretched out this one problem to fill 15 minutes... Instead of filling the video with so much fluff why not just work out more examples? It would be more helpful and less boring and I wouldn't have to skip over so much content.
Many don’t have a clue as how to do this For those who do ,yeh it gets long and tedious .There are things you have to know to do this problem.
Math Rebuild Corse how far does it go?? past Algebra 2? Trig up to and including pre- Calculus I am good in up to Algebra 1 and parts of Algebra 2 and parts of Trig
Definitely need help in pre- calculus and of course Calculus I see you don’t offer anything in Calculus 1-4
I agree. I stopped when he referred to math being taught in a boring manner.
In competative exams, this method is much worst.....
Give short cut to use it in competative examinations so candidates can solve in couple of seconds......
Great video!
Knowing the rules is key
With all respect for a video maker, mathematicians will never indulge in unnecessary manipulation. Video makers will.
Me neither
When would you use this?
This specific type of expression with these numbers? Who knows? Knowing how to evaluate something like this, however, is a good exercise in applying various mathematical rules. That way, if you ever find yourself working with some numbers or algebraic expressions, you will be able evaluate it correctly.
[(-4 * -4 * -4) / (-4 * -4)] = -4. next we take the reciprocal of (-4) and square it which gives of 1/16.
Super simple didn't need multiple choice, a calculator, or anything other than my brain and me typing into this comment.
I didn’t know about negative powers. So took a minute to check before watching. Naturally I then searched for the bit that showed whether my answer was correct 🤓 so I don’t need to watch the rest of the vid and move on to next puzzle
Work from the inside out good basic problem
exp -2 already pointed out towards answer a)
Yes, I got the answer of 1/16 in about 10 seconds in my head. But I have to comment that -4^3 does equal (-4)^3 so why make the distinction? Yes, I know that technically -4^3 really means an expression of (-1)*4^3 without the parenthesizes and not -4*-4*-4 but they are equivalent. You can just rewrite the expression to (-1)*4*(-1)*4*(-1)*4 and factor out the three -1's to (-1) to get (-1)*4^3. So why the distinction because they are equivalent? I solved the insides by using the formula a^c/a^b = a^(c-b) where a =-4, c=3 and b=2 for (-4)^1 by adding exponents then [-4^1]^-2 = 1/[-4^1]^2 and you multiply exponents to get 1/4^2 or 1/16. All you need is the laws of exponents to solve this problem in many ways.
4:14 starts
c)16
Just managed it in my head....Now i'm off to lie down lol.
Edit: My method. I used power rules. Because you're working with the same base (-4). So, you're left with -4^1. Anything to the -2 power = 1/ anything^2. Therefore 1/-4^2 = 1/16.
Oops, I just died on The Oregon Trail of math forgotten rules.
Great job sir.
Somebody must be interested in this.
Ah, the brackets. :)
-4. -64÷16=-4
i learned something on this one
Who invented that formula in the end?
Skip ahead to 4:10.
At the title card, I'm going with a> 1.16.
I will die on this hill, but -4 is a singular integer, just like positive 4. "-4" is not a compound expression of (-1 * 4). If algebra wants to separate -1 from 4, then you should write it as -(4), and that's when you use parentheses, not the reverse.
Sorry but in mathematics ... The exponent has higher *precedence* over the *unary* negative sign. It is the negative of ... 4 raised to the 3rd power.
Don't die on that hill. Established convention is 100% against you and for good reason.
There are similar constructs where the - symbol cannot be anything other than a negation operator, for example -(2+2)² and -x², and so for consistency the - symbol in an expression like -4² is treated as a negation operator too.
I had forgotten it already 🙂 though... I'm 64.
Thank you
Thanks sir
Thanks too mach
Of course, the formula. Next time I should be able to remember it. :(
b)1/16
16
Correct answer is (e) -4
C 16
1/16
Got it, even though I did the cubed 4 incorrectly.😊
I had completely forgotten the negative exponents part! I enjoyed the explanation very much.
A =1/16
It’s MATHS not math….
Would you be kind enough to explain why the numerator is not considered "negative four cubed" but is "four cubed to the minus"? To me, seeing a number written as "-4" is "minus 4", aka "negative 4", and adding the exponent to it does not mean (to me) to raise the number 4 to the power without its sign, since the sign itself is part of the identity of the value. Without that sign it becomes an absolute value, which is different from what is written and inferred by what was written.
At least to me, anyway.
Can you explain it to me so I understand, please? I'm not a dummy, but it just doesn't make sense.
He kinda mentioned it in regards to the order of operations. The exponent applies to the number. You could also think of it as (-1)*(4)^3. Consider how you would evaluate 0-4^3 vs 0-(-4)^3.
I like your explanation. I apologize but this is a little wordy.
(To simplify, I've left out the ^3.) Thinking of it as (-1)*(4) kinda makes sense, but not 0-4 since that changes it from me seeing the stated value of "negative four" to an expression that includes subtraction that is not in the original expression. Saying that it's "negative one (times) four" makes a bit more sense, however that also changes it by introducing multiplication that I did not see in the original expression.
What I saw when I evaluated the original problem as written (and got the correct answer) was I envisioned the number 4 having two possible addresses. One is the positive value that is four units to the right of Zero on the number line and the other is four units to the left of Zero. To denote the positive one, we simply write "4" without the plus symbol, but to denote the one on the left we write "-4" and call it negative four, aka minus four. Writing -4 is not a mathematical expression on its own like 0-4 or -1*4 are. It's the value of a single number with the minus symbol telling you that the digit resides four units to the left of Zero. In order to write "(-1)*(4)" so as not to use negation on its own, one would have to write "(0-1)*(4)", and the original problem would have to be written out "(0-1)*(4^3)", but that undoes the whole reason for the video, which is PEMDAS, as well as some good entertainment and intelligent discussion.
So, when I saw the original problem, his parentheses around the negative four on the denominator seemed optional to me. To calculate the numerator, it then became (-4)*(-4)*(-4). However, since I saw the denominator with the same base, I simply subtracted 2 from 3 and arrived at -4 for the new numerator and 1 for the new denominator. Then all I had to deal with was the negative power of the new expression of -4, and the only "math" I had to do was to subtract 2 from 3.
If the original problem had had -4^2 in the numerator inside the square brackets, my final answer would have been negative 1/64 since the final value of the numerator would have been 16, not -16, making it equal to the denominator (in my estimation). And this is where the whole rub comes in and where it doesn't seem logical to me.
I hope I've explained it clearly and without insults. @@MadSlantedPowers
Considering those last two expressions:
0-4^3 would be 4 cubed, and then subtract that from zero. The result would be -64.
0-(-4)^3 would be the quantity (negative 4) cubed that is then subtracted from zero. The result would be 64.
The result of -64 is as I mentioned in my long-winded comment previously: a numerical value that lives at an address 64 units to the left of Zero. To me, the number is known as "negative 64", not "zero minus 64" by the way it is written. That is also how I consume it mathematically and logically. If I'm wrong, I'll accept that.
This goes back to a similar video by Presh Tawalker. In those comments, it all boils down to the human just needing to be more careful about writing the expression in the first place. I write computer code throughout the day, and at times it includes formulas in both code, queries, and Excel cells. I'm OCD in being careful about not being lazy when writing my expressions because I don't want anyone (the computer or the next developer) to have any chance to misinterpret what is meant by my expression. In this way, I am certain to get the same result every time, no matter what. Relying on PEMDAS is for humans, not developers. That said, I'll rest my case and accept that, as a human, I may be mistaken in what I was taught. But as a developer, I am not. I write my code so as to be 100% crystal clear to prevent just such excursions and wasted time tracking down logic errors, which cost more time than most other errors do.@@MadSlantedPowers
@@hotflashfoto I guess I would consider -4^3 to be the opposite of 4^3. It's been a while since I wrote any code, but I have used Excel formulas and calculators. Right now on my home computer using NeoOffice, I entered "=-4^2" and got 16, so it seemed to interpret it as (-4)^2. I'll have to see what Excel on my work computer does. Using the calculator on my iPhone, I entered the following four keys: [-][4][x^2][=] and get -16 as a result. Likewise with entering it into Google or Wolfram Alpha.
You needn't continue down the Rabbit Hole if you don't wish to. Your evidence proves that computers need the extra steps to force them into a single result at all times. This video is a good example of teaching PEMDAS for humans, and yet our little discussion has shown that even with this, there are rules that need to be discussed.
I've jokingly told people that I don't have OCD, I have CDO, which is OCD alphabetized. In this instance you can see how it might apply. I coupled the learning from my childhood with the logic from my daily job and found a small blemish that I tore into a wormhole, so to speak. But in reality, all it is is a difference in how I approached the problem from how others approached it, including the presenter himself. I would consider it a win that we've both maintained our dignity and shown others how to communicate without being rude, as we see so often.@@MadSlantedPowers
Im naming my son Pemdas
A
Got it with a calculator. I forgot the rule when the exponent is a negative. Now let's see what I am missing. ;)
My name is Many and my answer is 1/16
I do have a concern with your answer. My answer is -1/16, this is why. When you have a base to a power to a power you need to solve the exponential part first before applied to the base (math algebra rule, MIT). For instance. 3^3^3 IS NOT 27 to the 3rd, (27^3), it is 3 to the 27th power. 3^27, so if you follow this well established rule then the answer is -1/16. Your comments, thanks.
I think in this case, brackets and parentheses take precedence due to order of operations. In your example, you are evaluating the exponents from top to bottom. However, if you wrote it as (3^3)^3, then the answer would be 27^3. I think all of his steps are correct.
You are talking about the case where an exponent is not presented as a single number, but is presented as a mathematical expression that itself includes an exponent.
That is not the case here. All the exponents are simply single numbers.
In the numerator the exponent is 3, and the base is 4.
In the denominator the exponent is 2 and the base is -4
And for the final part of the calculation, the exponent is -2 and the base is the value of the expression ( -4³ / (-4)² ).
This should also make it clear why, before even doing any calculation, we can see that -1/16 cannot be the answer. Ultimately we are calculating (something) to the power of -2, and (anything) to the power of -2 is ≥ 0.
does
-4^2 = (-4)^2
???
your first expression would cause confusion to other people. a math person would write it out more clearly
@@MartinHermans-dw3is and -4^3?
@@tomtke7351i changed my mind. The answer to your question is no, they are not equal
@@MartinHermans-dw3is so
"P" of PEMDAS should be employed profusely, right?
it entangles my mind to encounter
-4^2 now knowing it = -16 NOT 16.
thus E of PEMDAS over rules what else of PEMDAS? The."-" of -4^2 is NOT S of PEMDAS. What is it?
Review 5:50 - 7:12 the teacher explains this in detail. Example -4^2 means positive 4 times pos 4 then take the neg of that result. The - sign is NOT attached to the 4 UNLESS the -4 is enclosed in ( ) when using powers
1/16
a
Unnecessarily long and complicated indeed. The absolute sign means that the outcome is positive no matter what. Hence we only must calculate 4^3 / 4^2 = 4 and 4^-2 = 1/16. I did it in less than ten seconds.
There's no absolute value sign here.
But we can immediately rule out the possibility of a negative answer, because we are calculating (something) to the power of -2, and the power of -2 can not yield a negative result.
should have factored out the -1 and cancelled like a sensible person.
I used to be told that time is important in maths. Just do the steps and we'll be less confused. Talking too much may actually confuse the students more than step by step maths.
I started tweaking bro, u stretch the video sm
Dont blame the man for talking too much because this is his only chance to talk. At home his wife does not even allow him to even get a word in
I got 1/16 from the title screen
Nightmare explanation. Basics are disguised under an avalanche of words.
Too much talk.
Am I supposed to remember all this….😳
5 seconds in my head
@@thenetsurferboy sure buddy LOL
@@thenetsurferboy, I took a little longer, but it’s been a long time since I studied this. Doing math in my head sure helps as I get older.
He talks too much and promotes his channel and himself too much. This video can easily be five minutes
B-16
SHOCKINGLY VERBOSE WTF DO WE NEED A CALCULATOR FOR? LOL MOST AMERICANS ARE GOING TO (NOT WILL) MESS THIS UP
FOR CLARITY LMAOWRITE -4 AS (-1x4) YOU ARE GOING TO FRIGHTEN MOST PEOPLE WITH THIS SPORIOUS SPIEL
-16
Interesting channel. I like the problems but the explanations are way... too... slow... Sorry, I understand you're targeting school age kids but I look at them as a fun refresher and I can't stand watching that long. He's driving me a bit nuts actually.
Man, you really milk it. This 16 minutes video could have been done in less than two minutes .
16
A
a
1/16
A
a
1/16
1/16
1/16
1/16
1/16