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Rigorously (and nicely) presented. I note from comments that some others start by "rationalizing the denominator," as I did. In other words, multiply by sqrt10/sqrt10. It gets you to the desired result, just as your approach did. My wife says I'm always rationalizing.
yes but more complicated and prone to mistakes. Hopefully the student is alert enough to see it is a fraction divided by a fraction then proceed from there.
Thank you for a nice question and a clear explanation. With respect, I would consider multiplying the top and bottom by √10, resulting in √(50/8) + √(80/5) = √(25/4) + √16 = 5/2 + 4 at the top and each to be divided by 10, resulting in 5/20+4/10. (That’s 1/4 + 4/10= 25% + 40%= 65% = 0.65 or 13/20. Just for discussion). Thank you
There’s a rule called no square roots in denominator if there’s a square root in denominator you can undo it by multiplying the numerator and denominator. Another thing all fractions can be simplified to decimals.
There is no such "rule"... it is a common practice because it looks nicer and usually easier to work with further if necessary but it certainly is not a rule. Also not all fractions can be simplified to decimals, as fraction can be irrational and that can at best be approximated to a decimal but is not acceptable as simplified form.
It's not always good to convert fractions to decimals. Very often, fractions can cancel out. People who rely heavily on calculators will want to convert, but that often involves more work than just sticking with fractions.
Yes, this clears all the fractions within the fractions. And, in this problem, happens to clear ALL the radicals, too! Numerator: N = √40 [ √(8/5) + √(5/8) ] = √[40(8/5)] + √[40(5/8)] = √(8*8) + √(5*5) = 8 + 5 = 13 Denominator: D = √40 √10 = √(40*10) = √400 = 20 So, N/D = 13/20 or, = 0.65
love how you made that divide by 10 a multiplication problem. That is exactly why PEMDAS should really be PEMA. Apply the dividing by fractions rule to get rid of division then understand the rules x - y is the same as -y + x and -x - y is the same as -y + -x and that gives you PEMA which IMO eliminates the controversy with PEMDAS and which comes first between multiplication and division. oh and if you love mnemonic's, then instead of Please Excuse My Dear Aunt Sally, you instead say Please Excuse My Attitude. (My students get a kick out of that one)
I don't know why most people have a mortal fear and dread of square roots. I went on a blind date once with a colleague from work and while we were waiting for the desert to be served, I waxed lyrical about the beauty and power of square roots. My lady date excused herself and repaired to the ladies ablution room to refresh her facade and never returned! The waiter, a po-faced chap who would have not looked out of place as an embalmer at an undertaker's, came to inform that my date had done a runner through the fire escape exit and the bill needed paying. I was bereft, broken and beyond all consolation. Was I dejected? No. Was I despondent? No. Was I disappointed? A little. My date ran off with a waiter in tight trousers some months later and ended up running a hacienda and donkey sanctuary in Italy. I need a hug or some more questions like these gems from Math Queen. Thank you!
Lets make this a bit cleaner by turning the division into multiplication and since I am using a computer I will write the square root in exponent form which is 1/2 in this case (5/8 * 1/10) ^ 1/2 + (8/5 * 1/10) ^ 1/2 5/80 can simplify to 1/16 and 8/50 can to 4/25 so we now have (1/16) ^ 1/2 + (4/25) ^ 1/2 square root of 1 is 1 and square root of 16 is 4 so that is 1/4 square root of 4 is 2 and the square root of 25 is 5 so that is 2/5 2/5 + 1/4 has the common denominator of 20 2 * 4 over 5 * 4 is 8/20 1 * 5 over 4 * 5 is 5/20 5/20 + 8/20 is 13/20 No variable to prove and all work is shown so the final Answer is 13/20
I get why it might feel that way, but honestly, choosing to learn something new is a great use of time. That’s way more productive than most Saturday nights. 😉 I’m glad you found it interesting-thanks for watching!
I recently found in another RUclips puzzle that the square of x + 1/x for any x is x^2 + 1/x^2 + 2. That pattern jumped out for this numerator. Square of the whole fraction is (5/8 + 8/5 + 2)/10 = ((25 + 64 + 80)/40)/10 = 169/400. --> SQRT(169/400) = 13/20. (This is a variation on kareolausen819's solution, but without all the confusing square roots.)
This is a trivial problem to solve if you start by just squaring the entire fraction to get rid of the radicals, and then take the square root of the result.
you forgot the opening bracket AND a full bracket in the denominator on the line = (√5*√5 + √8*√8)/√5*√8] : √2*√5 it should be = [(√5*√5 + √8*√8)/(√5*√8)] : √2*√5 The subsequent lines are also missing the same full bracket. Otherwise the calculation result would be VERY different.
Well... I noticed a pattern with Fibonacci Numbers. It's not what you did. It's a little different. I guess I can say it is derived from this question. f:=n-> (((1+sqrt(5))/2)^n-((1-sqrt(5))/2)^n)/sqrt(5) (sqrt(f(n)/f(n + 1)) + sqrt(f(n + 1)/f(n)))/sqrt(f(n)*f(n + 1)) = f(n+2)/(f(n)*f(n+1)) And it works in non-integer values. Which is even cooler. * Oh, I see it now. I'll just leave it the way it is. Let you all figure it out.
Hey math friends! If you’re enjoying this video, could you double-check that you’ve liked it and subscribed to the channel? It’s a simple equation: your support + my passion = more great content! Thanks for helping me keep this going - you’re the best!
Not as hot as you !!!!!
Very nice explanation. Many other math problem videos I have seen are tediously long. You did it just right.
I always get in a good mood when this excellent teacher gives a lesson 😀
Nice!
I like both, the math and the well-structured explanation in fluent English 😊
🙂👻
Rigorously (and nicely) presented. I note from comments that some others start by "rationalizing the denominator," as I did. In other words, multiply by sqrt10/sqrt10. It gets you to the desired result, just as your approach did. My wife says I'm always rationalizing.
yes but more complicated and prone to mistakes. Hopefully the student is alert enough to see it is a fraction divided by a fraction then proceed from there.
Put everything on a common denominator: Big fraction = (√5*√5 +√8•√8)/(√8•√5•√10) = (5+8)/√400 = 13/20.
That’s how it is supposed to be solved.
Thanks for the fun and instructive videos! What’s the name of the app you’re using?
Thank you for a nice question and a clear explanation. With respect, I would consider multiplying the top and bottom by √10, resulting in √(50/8) + √(80/5) = √(25/4) + √16 = 5/2 + 4 at the top and each to be divided by 10, resulting in 5/20+4/10.
(That’s 1/4 + 4/10= 25% + 40%= 65% = 0.65 or 13/20. Just for discussion). Thank you
That's close to the way I did it.
There’s a rule called no square roots in denominator if there’s a square root in denominator you can undo it by multiplying the numerator and denominator. Another thing all fractions can be simplified to decimals.
There is no such "rule"... it is a common practice because it looks nicer and usually easier to work with further if necessary but it certainly is not a rule. Also not all fractions can be simplified to decimals, as fraction can be irrational and that can at best be approximated to a decimal but is not acceptable as simplified form.
It's not always good to convert fractions to decimals. Very often, fractions can cancel out. People who rely heavily on calculators will want to convert, but that often involves more work than just sticking with fractions.
Solution:
√(5/8) = √5/√8 = (√5 * √5) / (√8 * √5) = 5 / √(8 * 5) = 5 / √40
√(8/5) = √8/√5 = (√8 * √8) / (√5 * √8) = 8 / √(5 * 8) = 8 / √40
So the Numerator of the whole term is 5/√40 + 8/√40 = 13/√40
This means the entire term can be rearranged to:
13 / (√40 * √10)
= 13 / √400
= 13/20
You are my hero. I love your solutions.
Deutsche Susi oder englische Susi deine Kommentare sind immer mega useful.
LG Gerald
If you don't want to mess with fractions, simply multiply top and bottom by square root of 40 to get (8 + 5)/20.
If you square the expression, you get 169/400. The thing you learn from this is that (√(a/b)+√(b/a))^2=(a+b)^2/(ab). Pretty cool.
I just found it easier to multiply top and bottom the square root of 10. Then you get (5/2 + 4) /10. Multiply top and bottom by 2 to get 13/20.
How about just mult numerator and denominator by sqrt 40?
Simpler.
Yes, this clears all the fractions within the fractions. And, in this problem, happens to clear ALL the radicals, too!
Numerator:
N = √40 [ √(8/5) + √(5/8) ]
= √[40(8/5)] + √[40(5/8)]
= √(8*8) + √(5*5)
= 8 + 5
= 13
Denominator:
D = √40 √10
= √(40*10)
= √400
= 20
So, N/D = 13/20
or, = 0.65
Yeah, also what I did
You can multiply numerator and denominator by sqrt of 10, it is enough.
You are the math queen.
Soon to be the math dowager.
Great 👍 👌
love how you made that divide by 10 a multiplication problem. That is exactly why PEMDAS should really be PEMA. Apply the dividing by fractions rule to get rid of division then understand the rules x - y is the same as -y + x and -x - y is the same as -y + -x and that gives you PEMA which IMO eliminates the controversy with PEMDAS and which comes first between multiplication and division.
oh and if you love mnemonic's, then instead of Please Excuse My Dear Aunt Sally, you instead say Please Excuse My Attitude. (My students get a kick out of that one)
For me, the tricky part is remembering all the rules. What's the best way to remember when a rule should or shouldn't be applied?
Unfortunately, it just takes practice, but you can do it! I hope my videos help you over time 😉
PRACTICE PRACTICE PRACTICE 😁
I don't know why most people have a mortal fear and dread of square roots. I went on a blind date once with a colleague from work and while we were waiting for the desert to be served, I waxed lyrical about the beauty and power of square roots. My lady date excused herself and repaired to the ladies ablution room to refresh her facade and never returned!
The waiter, a po-faced chap who would have not looked out of place as an embalmer at an undertaker's, came to inform that my date had done a runner through the fire escape exit and the bill needed paying.
I was bereft, broken and beyond all consolation. Was I dejected? No. Was I despondent? No. Was I disappointed? A little.
My date ran off with a waiter in tight trousers some months later and ended up running a hacienda and donkey sanctuary in Italy.
I need a hug or some more questions like these gems from Math Queen. Thank you!
Lets make this a bit cleaner by turning the division into multiplication and since I am using a computer I will write the square root in exponent form which is 1/2 in this case
(5/8 * 1/10) ^ 1/2 + (8/5 * 1/10) ^ 1/2
5/80 can simplify to 1/16 and 8/50 can to 4/25
so we now have
(1/16) ^ 1/2 + (4/25) ^ 1/2
square root of 1 is 1 and square root of 16 is 4 so that is 1/4
square root of 4 is 2 and the square root of 25 is 5 so that is 2/5
2/5 + 1/4 has the common denominator of 20
2 * 4 over 5 * 4 is 8/20
1 * 5 over 4 * 5 is 5/20
5/20 + 8/20 is 13/20
No variable to prove and all work is shown so the final Answer is 13/20
How sad , I'm watching this on a Saturday night. Still very much interesting. Thanks
I get why it might feel that way, but honestly, choosing to learn something new is a great use of time. That’s way more productive than most Saturday nights. 😉 I’m glad you found it interesting-thanks for watching!
@@MathQueenSusanneu do tutoring ?
Thank you. Very clear.
Thanks a lot for watching ❤️
I recently found in another RUclips puzzle that the square of x + 1/x for any x is x^2 + 1/x^2 + 2. That pattern jumped out for this numerator. Square of the whole fraction is (5/8 + 8/5 + 2)/10 = ((25 + 64 + 80)/40)/10 = 169/400. --> SQRT(169/400) = 13/20. (This is a variation on kareolausen819's solution, but without all the confusing square roots.)
Got the same result by squaring the fraction, simplify to 169/400 then take the square root of that
(Sqrt[5/8]+Sqrt[8/5])/Sqrt[10]=0.65=13/20 final answer You find Square roots of fractions by separating them.
(Sqrt(5)/sqrt(8) + sqrt(8)/sqrt(5)) / sqrt(10)
(5/sqrt(40) + 8/sqrt(40)) / sqrt(10)
(13/sqrt(40)) * (1/sqrt(10))
13/sqrt(400)
13/20
Awesome ❤
This is a trivial problem to solve if you start by just squaring the entire fraction to get rid of the radicals, and then take the square root of the result.
It's not trivial to (correctly) square the numerator.
@ddichny Which part of squaring the numerator do you think is not trivial?
This looks tricky. It's a complex fraction. You'd start by simplifying the numerator.
My way of solution ▶
[√5/8 + √8/5 ]/ √10
= [(√5/√8 + √8/√5] /√10
= (√5*√5 + √8*√8)/√5*√8] : √2*√5
= [(5+8)/√5*√2²2] : √2*√5
= [13/√5*2√2] : √2*√5
= 13/√5*2√2*√2*√5
= 13/(5*2*2)
= 13/20 ✅
or
= 0,65
you forgot the opening bracket AND a full bracket in the denominator on the line
= (√5*√5 + √8*√8)/√5*√8] : √2*√5
it should be
= [(√5*√5 + √8*√8)/(√5*√8)] : √2*√5
The subsequent lines are also missing the same full bracket. Otherwise the calculation result would be VERY different.
Amazing... :)
( √(5/8) + √(8/5) ) / √10 =
... Note: 5/8 = 40/64 , 8/5 = 40/25 ...
= ( √(40/64) + √(40/25) ) / √10
= ( (√40)/8 + (√40)/5 ) / √10
= ( 1/8 + 1/5 )*(√40) / √10
= ( 5/40 + 8/40 )*(√40) / √10
= ( 13/40 )*(√40) / √10
= ( 13/40 )* √(40/10)
= ( 13/40 )* √(4)
= ( 13/40 )* 2
= 13/20
= 0.65
Got 13/20 with a shorter path, all by heart
Faster to multiply by square root of 40 both numerator and denominator.
Well... I noticed a pattern with Fibonacci Numbers. It's not what you did. It's a little different. I guess I can say it is derived from this question.
f:=n-> (((1+sqrt(5))/2)^n-((1-sqrt(5))/2)^n)/sqrt(5)
(sqrt(f(n)/f(n + 1)) + sqrt(f(n + 1)/f(n)))/sqrt(f(n)*f(n + 1)) = f(n+2)/(f(n)*f(n+1))
And it works in non-integer values. Which is even cooler.
* Oh, I see it now. I'll just leave it the way it is. Let you all figure it out.
{25}+{64}/{400}=5+8/20=13/20
It’s easier to work upstairs first, then ten comes at the end.
❤️🚀
❤ Dankeschön ❤
13
----
2
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10
I did it the hard way and got (2.5+4)/10= 0.65
13/20 in no sec
Square it and get nice simplifications
13/2=6.5 so 13/20=0.65
My XL spreadsheet says 0.65. We have tools, use them
🙃
Someone needs a date 😂
Love a smart, bubbly, beautiful woman!
красотка, целую в щечки.
.65? Pre video
No.
Too easy. A couple ways to solve.