Rewriting square root of fractionYouTube↗  IA↗(long shot)
Education  190.8K views 86 likes 9 dislikes   11 Feb 2016
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- [Voiceover] So we have here the square root, the principal root, of one two-hundredth. And what I want to do is simplify this.
Tak máme tady druhou odmocninu z (1 lomeno 200) a chci ji zjednodušit.
When I say "simplify it" I really mean,
Když říkám "zjednodušit", ve skutečnosti hledám druhou mocninu celého čísla,
if there's any perfect squares here that I can factor out to take it out from under the radical.
kterou bych mohl vytknout z výrazu pod odmocninou.
And so I encourage you to pause the video and see if you can do that.
Pozastavte si teď video a zkuste si to sami.
Alright so there's a couple of ways that
Dobře, takže máme několik způsobů, jak na to můžeme jít.
you could approach this. One way is to say, Well this is going to be the same thing as the square root of one over the square root of 200.
Jeden způsob je říct, že to bude totéž co (odmocnina z 1) lomeno (odmocnina z 200).
The square root of one is just one over the square root of 200.
Odmocnina z 1 je 1, lomeno odmocninou z 200.
And there's a couple of ways to try to simplify the square root of 200.
A máme několik možností, jak zjednodušit odmocninu z 200.
I'll do it a couple of ways here.
Zkusím to teď několika způsoby.
Square root of 200.
Odmocnina z 200.
You could realize that, OK, look 100 is a perfect square.
Mohli bychom si říct, že 100 je druhou mocninou celého čísla a zároveň dělí 200.
And it goes into 200. So this is the same thing as two times 100.
Takže je to totéž co dvojnásobek 100.
And so the square root of 200 is the square root of two times 100, which is the same thing as the square root of two times the square root of 100.
Proto je odmocnina z 200 stejná jako odmocnina z (2 krát 100), což je totožné s (odmocnina ze 2) krát (odmocnina ze 100).
And we know that the square root of 100 is 10.
A víme, že odmocnina ze 100 je 10.
So it's the square root of two times 10 or we could write this as 10 square roots of two.
Takže (odmocnina ze 2) krát 10 nebo taky 10 odmocnin ze 2.
That's one way to approach it.
To je jeden způsob.
But if it didn't jump out at you immediately that you have this large perfect square that is a factor of 200,
Pokud vás ale hned nenapadlo, že tak velká mocnina celého čísla dělí 200,
you could just start with small numbers.
mohli jste začít s malými čísly.
You could say, alright, let me do this alternate method in a different color.
Ukážu teď v jiné barvě, co jste mohli udělat.
You could say, ah it's the same color that
Mohli jste si říct…
I've been doing before.
Jejda, to je zas ta samá barva.
(laughing) You could say that the square root of 200, say
Mohli jste si říct, že odmocnina z 200… Že 200 je dělitelné 2.
Well it's divisible by two. So it's two times 100.
Takže je to 2 krát 100.
And if 100 didn't jump out at you as a perfect square,
No a pokud jste si neuvědomili, že 100 je druhá mocnina 10,
you could say, Well that's just going to be two times 50.
mohli jste si říct, že to je jako 2 krát 50.
Well I can still divide two into that.
A ještě můžu podělit 2.
That's two times 25.
To je 2 krát 25.
Let's see, and 25, if that doesn't jump out at
Když si ani u 25 neuvědomíte, že se jedná o druhou mocninu celého čísla,
you as a perfect square, you could say that that's not divisible by two, not divisible by three, four, but it is divisible by five.
můžete si říct, že to není dělitelné 2 ani 3 ani 4, ale 5.
That is five times five.
To je 5 krát 5.
And to identify the perfect squares you would say,
Druhé mocniny celých čísel najdeme tak, že si uvědomíme,
Alright, are there any factors where I have at least two of them?
jestli máme nějaké dělitele, které se v rozkladu objevují alespoň dvakrát.
Well I have two times two here.
No mám tady 2 krát 2.
And I also have five times five here.
Taky tu mám 5 krát 5.
So I can rewrite the square root of 200 as being equal to the square root of two times two.
Takže můžu odmocninu z 200 přepsat jako odmocninu z (2 krát 2…
Let me just write it all out.
Celé to rozepíšu.
Actually I think I'm going to run out of space.
Vlastně už mi dochází místo.
So the square root, give myself more space under
Takže ta odmocnina, udělám si místo,
the radical, square root of two times two times five times five times two.
odmocnina z (2 krát 2 krát 5 krát 5 krát 2).
And I wrote it in this order so you can see the perfect squares here.
Zapsal jsem to v tomto pořadí, abyste viděli ty dvojice.
Well this is going to be the same thing as the square root of two times two.
Bude to totéž co odmocnina z (2 krát 2…
This second method is a little bit more monotonous, but hopefully you see that it works,
Tato druhá metoda je trochu monotónní, ale doufám, že vidíte, že funguje.
(laughing) I guess is one way to think about it.
Tak si to alespoň můžete říct...
And they really, they boil down to the same method. We're still going to get to the same answer.
Stejně jsou obě vlastně totéž a dojdeme k témuž výsledku.
So square root of two times two times the square root times the square root of five times five, times the square root of two.
Takže (odmocnina z (2 krát 2)) krát (odmocnina z (5 krát 5)) krát (odmocnina ze 2).
Well the square root of two times two is just going to be,
První dvojice, to mi dá… To je prostě 2.
this is just two. Square root of five times five, well that's just going to be five.
Odmocnina z (5 krát 5), to bude 5.
So you have two times five times the square root of two, which is 10 times the square root of two.
Takže máme 2 krát 5 krát odmocnina ze 2, což je 10 odmocnin ze 2.
So this right over here, square root of 200, we can rewrite as 10 square roots of two.
Takže tohle, odmocnina z 200, se přepíše jako 10 odmocnin ze 2.
So this is going to be equal to one over 10 square roots of two.
Tohle pak bude rovno 1 lomeno (10 odmocnin ze 2).
Now some people don't like having a
Někdo nemá rád odmocninu ve jmenovateli,
radical in the denominator and if you wanted to get rid of that, you could multiply both the numerator and the denominator by the square root of two.
takže pokud se toho chcete zbavit, můžete rozšířit zlomek odmocninou ze 2.
'Cause notice we're just multiplying by one,
To vlastně jenom násobíme 1,
we're expressing one as square root of two over square root of two,
kterou zapisujeme jako (odmocnina ze 2) lomeno (odmocnina ze 2).
and then what that does is
A co získáme,
we rewrite this as the square root of two over 10 times the square root of two times
je (odmocnina ze 2) lomeno (10 krát (odmocnina ze 2) krát (odmocnina ze 2)).
the square root of two. Well the square root of two times the square root of two
Z těch odmocnin zbude jenom 2.
is just going to be two. So it's going to be 10 times two which is 20.
Takže máme 2 krát 10, což je 20.
So it could also be written like that.
Takže to můžeme zapsat také takto.
So hopefully you found that helpful.
Snad vám to pomůže.
In fact, even this one, you could write if you want to
Vlastně i toto by šlo napsat, zobrazit si to jinak.
visualize it slightly differently, you could view it as one twentieth
Mohli byste se na to podívat jako na (1 lomeno 20) krát (odmocnina ze 2).
times the square root of two. So these are all the same thing.
Ty jsou si všechny rovny.
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