Это видео недоступно.
Сожалеем об этом.
Square root of an exponent problem: what is sqrt(16^16)?
HTML-код
- Опубликовано: 3 янв 2024
- What is the square root of 16^16?
(A) 8^16 (B) 8^8 (C) 4^16 (D) 16^4
To simplify sqrt(16^16) we need to know the connections between radicals and the rules of exponents. I posted this question on my Instagram and it currently has over 1.2 million views. Here's my Instagram reel www.instagram....
Shop my math t-shirts & hoodies on Amazon: 👉 amzn.to/3qBeuw6
-----------------------------
I help students master the basics of math. You can show your support and help me create even better content by becoming a patron on Patreon 👉 / blackpenredpen . Every bit of support means the world to me and motivates me to keep bringing you the best math lessons out there! Thank you!
-----------------------------
#math #algebra #mathbasics
A smartphone is now $500 after a 20% discount. What was its original price?
Answer here: ruclips.net/video/IYmiFQUsb2Q/видео.html
500÷.8 gives $625
625
20% discount means that it is now 80% of its original price. Let "p" be the value for the original price and we can write the equation 0.8p=500 and p=$625
500 $ / 0,8 = 625 $
625
My first instinct when I saw the question was 16^8. When I didn’t see that answer, I had to think about it slightly longer, but saw 4^16 = (4^2)^8, therefore 4^16 had to be equivalent to 16^8.
Me too. I was about to say “none of the above”
Oh I forgot about the rule, thanks!
I thought the radical sign becomes /2 for the exponent and got 16^8 ( sqrt = ^1/2 ). Same value but unfortunately not a selectable answer
yeah it's correct too!!
Yeah this was the way my brain went and was confused when it wasn’t a choice. Glad to know it was correct and he didn’t somehow disprove that 😂
What is your problem?
It's 4^16 same thing
how is that unfortunate you literally can convert it to the c choice
Also viable solution is to consider 16¹⁶ as 16 × 16 ×16... and so on.
If you make this, you can ask square root of not 16¹⁶ but of all it's elements which is sqrt(16) × sqrt(16) × sqrt(16)...
And because we know that square root of 16 is 4 - that whole eqution is just 4 times itself 16 times or just 4¹⁶
This explains it perfectly for me. I understood the video but this made it easier to understand.
Here's my process. The square root is equivalent to dividing the exponent by 2, so we get 16^8. Based on that, we know it's not B or D, because if a^b=a^c, then b must be equal to c, and if a^b=c^b, then a and c must have at least the same absolute value (assuming that we're dealing with real numbers). B and D both contradict those rules, so neither of them can be correct.
That just leaves us with A and C. And with only two options, I decided I could just check each one separately. 16^8 is (2^4)^8, or 2^32. 8^16 is (2^3)^16, or 2^48, so the answer can't be A. 4^16, meanwhile, is (2^2)^16, or 2^32, which is what we're looking for.
Of course, now that I stop to think about it, I realize that 16^(16/2) can easily be reinterpreted as (16^(1/2))^16, so I could just take the square root of the base and be done with it... yeah, I have a bit of a one-track mind.
its okay i did the same thing lol
Just to add to the list: sqrt(x^m) = x^(m/2) as he did, to get 16^8. But then 16=4^2, so 16^8 = 4^2^8. a^b^c = a^(b*c), so 4^2^8 = 4^16
An example would be the square root of x^16, which is x^8
Theoretically, that would make it 16^8. Then with some algebraic manipulation via exponential law, the answer should be C.
Edit: I won't lie, I didn't know you can just put the exponent outside of the square root.
Literally same!
@@m.i.c.h.o (a^m)^n = a^mn (either m or n can be 1/2 or any other root)..
its likely the simples way of doing it... but thes another trick
a^m = (a^(1/2) )^m*2 .. if we put numbers in
sqrt(16^16) = sqrt(4^32) = 4^(32/2) = 4^16
= [2^32, 4^16, 16^8, 256^4, 65536^2, 4294967296]
I converted 16^8 into 2^32 then (2^2)^16 and done with 4^16:))
@@studiouskid1528 ..when u know how..yes... thats why teaching the how is super important...
@@studiouskid1528 It does take seconds. I'm just commenting my thought process... relax lol
I really needed that ending because I was CONVINCED that by the rules of exponents the answer was 16^8. Then by process of elimination C was the only answer that wasn't ridiculous. Now it all makes sense.
Yeah, I came up with 16^8 initially as well, but 16 = 4^2, so 16^8 = (4^2)^8 = 4^16.
16⁸ basically equates to 4¹⁶
@@callmeweeb4350 Not just basically, they are equivalent expressions, period.
I’m not a math person, but I’m trying to learn what I couldn’t appreciate in school. I know this isn’t what the question was asking, but from a practical computational math standpoint I’m glad that my assumption of resolving the exponent first, then sqrt’ing the result at least got an equivalent answer.
Really love ur calculus vids once I'm a new AP calc learner. Hope you upload more calc vids and enrich your viewers knowledge.
Thanks! Please visit “bprp calculus” for my calculus tutorials
@@bprpmathbasics Easiest way- 16^16=(4^2)^16=(4^16)^2. If you square root this, just get rid of the ^2 so you get 4^16.
Since (x^a)^b = x^(ab), and sqrt(x) = x^(1/2), sqrt(16^16) = 16^(16/2) = 16^8.
Of course because ab = ba, you can also say that sqrt(16^16) = (16^(1/2))^16, i.e. (sqrt(16))^16 = 4^16 - hence you can just take the exponent out from under the radical sign and put it on the outside.
And of course 4^16 is 16 lots of 4 multiplied together, but 16 lots of 4 is equal to 8 lots of (4*4), so 4^16 = 16^8.
So there are multiple routes to the same answer, which illustrates the underlying nature of exponentiation.
I did 16/2 (because it's the square root, which without the exponent would be x^(1/2)) making it 8, and just keeping the base 16. 16^8 is 4.294e9 whatever, the same as 4^16.
2v16^16 same as 16^8 and 4^16 (v being square root I don't have the notation)
It seems for direct than putting the exponent outside of the radical. Then, by inspection, it's clear that C is the answer.
basic rule: nroot(b^m) = b^(m/n) , so 16^8 and then make base smaller (root) by expanding (double) amount of multipliers 4^16 or 2^32.
Think you are correct.
Yeah, both of your methods are not great, though, cause you can't do that thing for the negative numbers. Like if it was sqrt(-16^16), you'll get sqrt(-16)^16, which would resolve only by including complex numbers.
So yeah, it's better to do something like:
16^16 = (4^2)^16 = (4^16)^2
And sqrt((4^16)^2) = 4^16.
Also can get it with 16^16 = 16^(2*8) = (16^8)^2 and sqrt of that is 16^8, since we don't see any answer like that - monitor the answers and see that 16^8 can be written as (4^2)^8 = 4^16.
Using your methods you have to specify that either "b" should not be negative, or "m" should be even, or both.
If I saw this on a test with multiple choice. I would do this. 16 to the power of 16, hit enter, then square root that number.
Then check each answer.
THus it would end up being answer C, 4^ 16, since its a different number.
If I had to show work and my calculator didn't function that high... my answer would be way different, and probably marked wrong. Since i would never end up with 4^16...
However there is a long answer form in all math examines. Doing the multiple choice first, mark questions you struggled to get like the one above in how I got hte anaswer, to any answers on teh written that are simliar you cna use these steps...
What we find is that all we need to do is the sxquare root of 16 and leave its exponent alone. THus 4^ 16. Which makes sense when you think about it.. since the base number is a determinant... for a subset function... exponeneneneneent...
That is how I did it...
BPRP, I love your videos, but I think the way you explaned the notes could be more precise if you clarify that sqrt(b^m) = sqrt(b)^m if b is non-negative.
Power rules are fun! As a challenge I made the question and the solutions into powers of 2, multipying and dividing. Kinda cute how very different the scale is on some of the solutions.
im in 8th standard and what i went with was underroot 16^16 =16^8 = (4^2)^8 = 4^2x8= 4^16
Edit: I was thinking it in a wrong way. As pointed in comments. Leaving the original comment for reference
Original:
I was assuming the problem √(16^16) will be represented as 16^16^1/2. So the answer would be 16^4.
Am not sure why √(b^m) would be considered (√b)^m. Could you explain this concept may be?
that is not how you represent it, √(16^16) = (16^16)^(1/2) and through exponential laws we can multiply the powers 16^(16*1/2) = 16^8, then from this answer we know that 16 is just 4^2, so we have (4^2)^8, apply same law, 4^(2*8) = 4^16, therefore the answer is C.
@@lolgalaxy4406 yeah completely forgot about exponent laws. Thanks. 😅
Easiest way- 16^16=(4^2)^16=(4^16)^2. If you square root this, just get rid of the ^2 so you get 4^16.
I happened to see this square root exchange rule in some Japanese textbook, sqrt(a^x)=(sqrt(a))^x
I transformed it to 16^8 then i saw it wasnt a solution. I then made it to this: 16^8= (4^2)^8=4^(2×8)=4^16
Ok, so you can also solve this with the strat of looking for simillar answers that is based on that wrong answers will be simillar to the real answer in order to throw you off
We have 8^16 ,8^8, 4^16 and 16^4
Since 8^8 is simillar only to the 8^16 we can remove it
Now, 8^16 is simillar only to the 4^16, so we would leave 16^4 out and vice versa. So the most likely answer is 4^16.
Now, if we square it then we will have (4^16)^2 or 4^32 or 16^16, so it's a correct answer.
Thank you for reading on how to find the right answer the wrong way.
I just interpreted this as (16^16)^(1/2) and then because of the powers of powers rule the exponent became (16/2) or 8. Exact same answer as 4^16. Is this just situational to this problem or could you do it like this always?
Edit: Just watched the whole video. Jumped the gun again lol.
i used deductive reasoning. basically sqrt(a^x)=sqrt(a)^x as far as something like this would go, therefore the answer is c
Who is the young lady at the beginning. Can she do the pen color switch effectively? Or is she still training?
For completeness and since all options given have bases that are powers of 2, some equivalent answers to 4^16 are: 16^8, 2^32, and 8^(32/3).
Completeness? This isn't a question of solving for a variable. There are infinitely many ways to represent a^b = x if a and b includes all real numbers. If you limit a and b to integers then the solutions are 2^32, 4^16, 16^8, 256^4, and 65536^2.
@@matta5749 The original options were given using bases of 4, 8, and 16. I simply stated what the exponents should have been if any of the other options were to be correct. I threw in the base 2 answer in case anyone were to read this and wonder how the 8 answer came to be.
Yeah my first guess was 16^8 as I did the blue notes way first but then I saw it wasn't an answer, so went back and remembered a^b^c = a^(b*c) and then rewrote 16 as 4^2 so 4^(2*16)^0.5 = 4^(2*16*0.5) = 4^16
So.. a sqrt is the same as ^1/2, so the question is (16^16)^(1/2). As it is possible to multiply such exponents, it is the same as (16)^(16 * 1/2) which gives 16^8. Not in the answers, so we split the base 16 to 4*4 which gives (4*4)^(8). 4*4 can be written as 4^2 which gives (4^2)^8 and we multiply the exponents again which gives finally (4)^(2*8) = 4^16 i.d. answer C.
imo it it way easier to realize that a root is just a specific type of power, a fractional power. A square root is just a power of 1/2 while a third root is a 1/3, etc. So that just equates to 16^((1/2)(16)) -> 16^(16/2) -> 16^8
This also allows you to easily see that square root can be on on either side, it does not matter which happens first.
I was learning about slope one day and the equation to calculate it is m= (x2-x1)/(y2-y1). I was wondering what could be the greatest value of m since it would undefined if it's a straight line.
m is not going to be undefined for a straight line
the equation for a straight line is y = mx + c , if m is slope
m = Δy/Δx = Tan(θ); (Delta y/Delta x), Delta means difference in math
Tan(zero difference between Δy and Δx) is zero angle not undefined
for a straight line Δy/Δx is just a way of calculating the Tangent angle (Slope), if its zero u can just ignore it (y = (0)x + c -> y=c (a line intersecting the y axis at c)
if f(x) = y = mx + c ,
the slope is
m = Δy/Δx = Tan(θ) = Sin(θ)/Cos(θ) = Cos( θ - (π/2) )/Sin( (θ - (π/2) ); θ in radians
-= f'(x) (the first derivate of mx + c, wich is equal to m)-
..if Δy is approaching infinity(or very large), and Δx is infinitassible small m will approach infinity
wich gives 0
In general: (a^b)^c = a^(bc) = a^(cb) = (a^c)^b
So (16^16)^0.5 = (16^0.5)^16 = 4^16 is the only answer that matches.
Okay my work before watching: sqrt(16^16)=sqrt(16^8*16^8)=16^8 since 16=2^4 16^8=(2^4)^8 by the rule of exponents this is equal to 2^(4*8)=2^32. So the correct answer must be expressible as 2^32. So the base must be equal to 2^n where n is a factor of 32. Since 32 is not divisible by 3 we can immediately eliminate A and B because 8=2^3. This leaves C, 4^16 and D, 16^4. So checking D first, 16=2^4, so 16^4=(2^4)^4=2^(4*4)=2^16. So D is incorrect. That leaves C, 4=2^2 so 4^16=(2^2)^16=2^(2*16)=2^32. So the correct answer is C.
Edit after watching: This was the thing that made me struggle with some basic forms of math in school. I always forgot the "simple methods" because I found it easier to memorize relationships rather than formula. This lead to me being extremely upset when I learned Euler's formula (e^ix=cos(x)+isin(x)) and realized that it would have made calculus III MUCH easier because I always forgot the half-angle formula during tests, but I never once forgot Euler's formula.
C) 4^16. Square root of 16^16 is square root of 2^64 which is 2^32, which is (2^2)^16
0:40 Okay, fine, if you insist: sqrt(16^16) = C: (4^16)
Step-by-step, here's how I broke it down: sqrt(16^16) = (16^16)^(1/2) = 16^(16/2) = 16^8. Wrong base, so I need to change the base. 16^8 = (4^2)^8 = 4^16. Therefore, the answer is C: (4^16).
3:57 I did solve like that, though. :)
I really don't like any of those answers, to be perfectly honest. I understand that C is correct, but a far more useful way of looking at the problem and knowing whether your students understand exponents in general would be 16^8. This is either a question hiding a second part (meaning it's a bad question, because we can't disambiguate which part of the question the student was unable to perform if they miss it and thus learn more as teachers) or they really want their students to be looking for situation where they can find the square root of the base instead of treating it as a 1/2 exponent. When you see sqrt(x^16) with that mindset, you can't do anything.
I was yelling at my screen that it should be 16^8, and I was seriously doubting myself because that wasn't an answer.
But I was right, I did remember that sqrt(x) = x^1/2.
This is why I think math is a poor fit for multiple choice questions. So many things can be written in a myriad of ways that are all correct.
C?
I was like DAMN A BPRP VIDEO I CAN DO BEFORE EXPLANATION and its math basics
My answer before watching the video is: I don't know and I'm not going to guess. I will just watch and learn.
The blue part is what I'd do, sqrt is just 1/2 power so it would be 16^8 which is the same as 4^16 but without a calculator I wouldn't know that to choose C.
Hi! I have a question: _where does factorials fall under order of operations like PEMDAS/BODMAS?_
For example, the expression:
_3n!_
If _n_ = 3 then would the answer be 18 or 362880?
You cannot treat it as (3*3)!
As you have given 3n! Where n=3
We usually do not use parentheses to separate variable and coefficients until they are not single terms. Such as you have to use parentheses if you want to write 3(n-1)! But for the sake of ease we don't use them when we have something like 3(n)!
Now coming to your question, the same thing applies, as the question is not (3n)! We have 3n! that is similar to saying 3(n)!
As factorial is a function, you are only plugging in 'n' if you don't use parentheses.
If you still don't understand you can reply back. I am actually not that good at explaining. Sorry for that.
@@_elusivex_ So _3n!_ can be written as:
3n! = 3 × n!
Then in that case, if _n_ = 3 then the answer would be:
3 × 3! = 18
Correct?
@@_Just_Another_Guyyup , factorials are always done first
You have written 3n! W
, so You first have to do n! Then multiply by 3
If you had (3n)! Then you would first do 3n then do its factorial
A rule you can think about with order of operations is that, the more advanced operation is done prior to a more elementary operation. Cheerful Calculations 🧮.
@@_Just_Another_Guy yeah. nice and keep learning
Over 40 years and I can't believe I still remember it.
no, I did just split 16^^16 in to 16^8 * 16^8, and square root both then multiply you would get 16^8=4^2^8=4^16
Another route to the solution that I expected to see was to write
4* 4 = 4^2 = 16
And therefore
sqrt(16^16) = [ 16 ^ (1/2)] ^ 16 = [ (4 ^ 2) ^ 1/2] ^ 16
Keep 4 as the base, multiply all the exponents, and voilà.
let's use an example that we can wrap our heads around and solve this problem in 3 steps: 4^9=262144 , sqrt(4^9)=512 , 2^9=512
sqrt(a^b)=(sqrt(a))^b
sqrt(16^16)=sqrt(16)^2
sqrt(16^16)=4^16. Answer: C 4^16
The answer is C: 4^16. To get to that answer, you start with sqrt(16^16). Another way of writing that is 16^(16/2). Simplified, it becomes 16^8. 16 can also be written as 4^2, so you have (4^2)^8, or 4^(2*8), or 4^16.
EDIT: His method was a lot simpler.
Ah i got 16^8 and was wondering for a sec since I forgot you could convert it like you said!
I didn't know this rule and I got 16^8... That immediately eliminated B and D as options. So then I would do 16^8 = X^16, knowing full well that answer had to be to the power to 16. Make that 8 Log 16 = 16 log X....Solve for X... = 4
Could you explain the mechanism behind your processing of the problem please? I understand that there are quick 'laws' that are shorthand ways of remembering what to do but I am interested in exactly **how** we can remove the exponent from root 16^16 and turn it into (root 16)^16. What is the mechanism behind that operation? Thanks in advance.
u made this easyquestion hard
I think it's important to mention that the trick at the start only works for positive bases.
Sqrt((-1)²) and (sqrt(-1))² are not the same.
"Forgetting" how to do something shows it's lack of actual importance. Words to live by! 😁
Forgot that there's also that way to do it and was wondering why 16^8 wasn't there. Thanks
WE MAKING IT OUT 7TH GRADE WITH THIS ONE!🗣️🗣️🗣️🔥🔥🔥
omg i was freaking out until you posted the note about m/2 as an exponent. i thought i had learned math wrong. i knew it wasn't one of the answer options, but when you didn't list it as a second way to solve at first before listing the don'ts i was nervous af.
Good stuff
I don't know why this isn't selectable because you just have to rewrite the equation in a more normal form to make it obvious. (16^16)^(1/2) = (16^(16*1/2)) = 16^8.
The square root of 16^16 is 16^8
This is not in the options, but D is obviously wronh
16 = 4^2
So the value is (4^2)^8 = 4^(2*8) = 4^16
The answer is C
nice formula
That works, but also sqrt(b^m) = b^(m/2)
Always look for perfect squares within the divisibility of the base of an exponent. I feel like the answer should really be 2^32 if you want to keep "simplifying" the base.
from the start i went with the other way getting 16^8 and i thought to myself: okay, this is not an answer BUT! I can fragmentize 16 into 4^2 and because (4^2)^8 is equal to 4^2*8 I quickly got the answer
C) 4^16
Just convert sqrt(16^16) to sqrt(4^2)^16 and cancel 2nd power with square root then 4^16 is answer
C, but I didn't know how to prove it. Thank you!
I did it like this
√16¹⁶
= √(4²)¹⁶
= √[(4¹⁶)²]
= (4¹⁶)²/²
= 4¹⁶
Unnecessarily complicated, I forgot the √(b⁵) = (√b)⁵ rule. Thx bprp
My initial instinct was 16^4, but I thought about it for a bit and got the right answer. Kinda embarrassing that I had to even think about it considering I have a math degree 😅
the answer is actually 16^8. check it.
when you think of what exactly a "square root" is - the number that, when multiplied by itself becomes the number in the square root box - it becomes obvious that you just need to reduce the times it is being multiplied by itself (in exponents) by half in order to attain the answer.
understanding the whys and hows can actually save you from doing complex maths unnecessarily. plus, i ALWAYS love penciling in an "e." option on tests ;)
Can you reduce 16^8 into one of the answer choices?
@@sasukesuite1 sure, but where's the fun in that 😄
4^3 = 64
Sqrt(64) = 8
Sqrt(4^3) = 2^3
Therefore Sqrt(x^y) = Sqrt(x)^y
And this is how I use trial and error to solve math problems in school.
(follow at your own risk)
since sqrt(x) = x^1/2, sqrt(16^16) = 16^16^1/2 or 16^16x1/2 which is 16^8. Fininshed.
I was confused as to why this wasn't in the answers but I later realized that 16^8=4^16 because 4^2=8 and 4^2^8= 4^2x8 or 4^16.
square root is power 1/2 so sqrt(16^16) = (16^16)^1/2 = 16^(16 * 1/2) = 16^8, the proposed choice are unnecessarily complicated
At some point in my life, I learnt to solve differential equations, but I managed to forget it, and even forget these rules around powers.
4^16
Funny thing is, the moment I saw the question I though 16^8 but as there was no option for that, I though I was wrong :)
So I got the right answer but a different version of it, sqr = x^1/2, and so sqr(16^16) = 16^16/2 = 16^8 and 16^8 = 4^16
This video made me realize that x^k*x^k is not only egal to x^(k+k) but also (x*x)^k because (x^2)^k = x^(2*k), ye..
The annoying part is that the answer can be written in multiple ways which can through off students if they don't do it that ONE way that gives the answer.
I got 16^8 as well. If sqrt(x) = x^1/2, then sqrt(x^x) = x^1/2×x = x^x/2.
It would be C or 16^8 but since that’s not an option the same thing is C
4×4 =16 or 4² equals 16. Not sure. Please someone correct me. 🎉 or 4...16 I don't have the symbol for it on my phone .it's letter c.and no one taught me that yet.
4×4 or 4^2 is 16 indeed. Just use ^ to mark exponentials.
@@mahousenshi00 thank you
isn't this video incompatible with the sqrt( -4^2 ) video where you compute the square before computing the root? otherwise the answer would have been:
2i ^ 2 = -4.
can you please explain?
It's valid, cause 2i^2 = (2^2)*(i^2) = 4*(-1) = -4
But it'd require using complex numbers, so this solution is not great.
was confused at the start as i wouldve just put 16^8
I got C
I just split 16^16 into 4^16 and 4^16, and moved it out of the square then I got my answer
For once I managed to think of all the right answers on youtube math video before watching it.
For people saying 16^8 ;;; Notice its the same as 4^16 since ;;; There are half as many 16s in 16*16*16*... and twice as many 4s in 4*4*4*... But every two 4s makes a 16
since something square root is to the power of 0.5 i thought it was (16^16)0.5 so 16*0.5 is 8 and i got 16^8
I also got 16^8. I was annoyed to see it not a choice. Took me a minute to figure out it was the same as 4^16. Could tell by looking at it that it wasn’t b or d. Would have to do some work to figure out why a doesn’t work.
i confused exponent with reciprocal and got 16^15 from (16^16)^-1. I think my brain needs to be turned off then on again
CAN you solve a^b=c^d 🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏
Guessing D. As sqrt equals ^.5.
I know the answer is 16^8 but I lowkey forgot how to change it to one of the answers.
I dont know why but i thought of 16^8
But felt stupid after not seeing that in the options
c 4^16
c?
Why is 2^32 not the right proposed answer ? It should be the final solution.
4^16, 16^8, and 2^32 are three different representations of the same number: 4,294,967,296, which is the answer to the problem.
@@129140163 You dodged the question.
When in doubt, pick C.
16^16/2 = (4^2)^16/2 = 4^16
I solved this in the most roundabout way possible.
I wasn't confident that I remembered all the rules for resolving radicals so i converted everything to powers of 2 and found the equivalence that way. I blame binary counting.
16^8
i first thought 16^8 and was confused as to why it wasnt an option
C.