Simplifying hairy exponent expressions Khan Academy English Czech

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- [Voiceover] So let's get some practice simplifying hairy expressions that have exponents in them.

Procvičme se ve zjednodušování takových děsivých výrazů s mocninami.

And so we have a hairy expression right over here

Tady máme děsivý výraz.

and I encourage you to pause the video and see if you can rewrite this in a simpler way.

Pozastavte video a sami zkuste, zda jej dokážete zjednodušit.

All right, let's work through this together

Dobrá, zkusme to společně.

and the first thing that jumps out at me

První věc, která mě upoutá, je čitatel zde.

is the numerator here. I have the number 125 raised to the 1/8 power times the same number, the same base, 125, raised to the 5/8 power.

Mám 125 na (-1 lomeno 8) krát to stejné číslo, tedy 125, na (5 lomeno 8).

So I can rewrite this numerator.

Mohu tedy upravit čitatel.

I can rewrite this numerator using what I know

Čitatel mohu přepsat za pomoci znalostí o vlastnostech mocnin,

of exponent properties as being equal to 125 to the sum of these two exponents.

tedy je roven 125 umocněno na součet těchto mocnin,

To the negative 1/8 power plus 5/8 power.

tedy umocněno na (-1 lomeno 8) plus (5 lomeno 8).

All of that is going to be over the existing denominator

To celé bude lomeno současným jmenovatelem,

we have, which is five to the 1/2.

který je 5 na (1 lomeno 2).

All of that is going to be over five to the 1/2 power.

To celé bude tedy lomeno 5 na (1 lomeno 2).

So, these are equivalent.

Všechny jsou ekvivalentní.

Notice, all I did is I added the exponents,

Vše, co jsem udělal, je, že jsem sečetl exponenty.

these two exponents because I had the same base

Tyto dva exponenty, protože mají stejný základ

and we were taking the product of both of these 125 to the negative 1/8 and 125 to the 5/8.

a my bereme součin 125 na (-1 lomeno 8) a 125 na (5 lomeno 8).

And so negative 1/8 plus 5/8, well that is 1/2.

(-1 lomeno 8) plus (5 lomeno 8), to je rovno (1 lomeno 2).

Tohle tedy je…

this right over here is, 1/2.

Tohle zde je (1 lomeno 2).

So, this is 125 to the 1/2, over five to the 1/2.

To je tedy 125 na (1 lomeno 2), to celé lomeno 5 na (1 lomeno 2).

Well, that's going to be the same thing.

To je to samé jako…

This is going to be equivalent to

To se bude rovnat…

125 over five, over five, to the 1/2 power. To the 1/2 power.

125 lomeno 5, to celé umocněné na (1 lomeno 2).

If I raise 125 to the 1/2 and I'm dividing

Vezmu-li 125 na (1 lomeno 2) a vydělím 5 na (1 lomeno 2).

by five to the 1/2, that's the same thing as doing the division first and then raising that to the 1/2 power.

To je stejné jako nejdříve vydělit a potom mocnit na (1 lomeno 2).

Well, what's 125 divided by five?

Kolik je 125 děleno 5?

Well, that's just 25.

To se rovná 25.

Now what's 25 to the 1/2?

Kolik je 25 na (1 lomeno 2)?

Well, that's the same thing as the principal square root of 25, which is equal to five.

To je to samé jako odmocnina z 25, což je rovno 5.

And we're all done, that simplified quite nicely.

A jsme hotovi, to se zjednodušovalo docela snadno.

Let's do another one of these.

Udělejme další podobný.

And this one is a little more interesting, 'cause we are starting to involve a variable,

Tento je poněkud zajímavější, protože jsme začali s proměnnými,

we have the variable w,

máme zde proměnnou ‚w‘,

but it's really going to be somewhat the same process.

ale ve skutečnosti to bude podobný proces.

And here the thing that jumps out at me is the denominator.

Hned na mě vyskočí jmenovatel.

I have the same base, 3w-squared.

Máme stejný základ 3 krát (w na 2).

3w-squared raised to one power, one exponent,

3 krát (w na 2) , umocněno na nějakou mocninu,

times the same base, 3w-squared, raised to another power.

krát ten samý základ 3 krát (w na 2) umocněn na jinou mocninu.

So, this is going to be equal to,

Tohle tedy bude rovno…

this is going to be equal to

Tohle bude rovno…

our numerator we can just rewrite it 12w to the seventh power over negative 3/2

Čitatel si můžeme přepsat na 12 krát ((w na 7) na -3/2) lomeno jmenovatel,

over our denominator we can write as this base, 3w-squared,

ten můžeme psát jako náš základ, 3 krát (w na 2),

and we can add these two exponents.

a můžeme sečíst naše dva exponenty.

So, we could add negative 2/3 to negative 5/6.

Sečteme (-2 lomeno 3) s (-5 lomeno 6).

Negative 2/3 to negative 5/6.

(-2 lomeno 3) s (-5 lomeno 6).

Well, what is that going to be?

So, let's see, if I do negative, negative 2/3 is the same thing as negative 4/6, minus 5/6,

Podívejme, (-2 lomeno 3) je to samé jako (-4 lomeno 6), minus (5 lomeno 6).

which is equal to negative 9/6, which is equal to negative 3/2.

To je rovno (-9 lomeno 6), což je rovno (-3 lomeno 2).

So, this right over here is the same thing as negative 3/2.

Tohle celé je tedy rovno (-3 lomeno 2).

Let me just write that. Negative 3/2 power.

Zapíšu to. Exponent je (-3 lomeno 2).

Negative 2/3 plus negative 5/6 is negative 3/2.

(-2 lomeno 3) plus (-5 lomeno 6) je rovno (-3 lomeno 2).

Now, what's interesting is I have a negative 3/2 up here and I have a negative 3/2 over here,

Zajímavé je, že mám (-3 lomeno 2) tady a mám to i tady,

so, we can do the same thing we did in the last problem.

můžeme tedy udělat to stejné co minule.

This could simplify to 12w, 12w to the seventh power, over 3w-squared.

To se zjednoduší na 12 krát (w na 7) lomeno 3 krát (w na 2),

3w-squared. All of that, all of that to the, all of that to the negative 3/2 power.

to celé umocněno na (-3 lomeno 2).

Notice what we did here.

I had something to the negative 3/2 divided by something else to the negative 3/2.

měl jsem něco na (-3 lomeno 2) děleno něco jiného na (-3 lomeno 2).

Well, that's the same thing as doing the division first and then raising that quotient to the negative 3/2.

To je to samé, jako bychom nejdříve dělili a až poté umocňovali na (-3 lomeno 2).

And what's nice about this, this is pretty straightforward to simplify.

Na tomto je pěkné, že to můžeme zcela přímočaře zjednodušit.

12 divided by three is four,

12 děleno 3 je 4,

and w to the seventh divided by w-squared,

w na 7 děleno w na 2,

well, we could divide both by w-squared

obě vydělíme w na 2,

or you could say this is the same thing as w to the seven-minus-two power.

nebo bychom mohli říct, že je to jako w umocněno na (7 minus 2).

So this is going to be w to the fifth power.

Tohle tedy bude w na 5,

W to the fifth power.

And so it all simplified to 4w to the fifth power to the negative 3/2. To the negative 3/2.

To celé se tedy zjednodušilo na 4 krát (w na 5), to celé umocněno na (-3 lomeno 2).

Now, if we want to, this is already pretty simple

Teď, pokud bychom chtěli… Tohle už je dost zjednodušené,

and at some point it becomes somewhat someone's opinion on which is expression

v určitých případech záleží na každém, co komu připadá jednodušší,

is simpler than another, and it might depend on what you're using the expression for,

může záležet i na tom, k čemu chcete výraz použít,

but one could argue that you could keep trying

někdo by ale mohl chtít, abyste pokračovali ve zjednodušování.

to simplify this, this is the same thing as, this is the same thing as four to the negative 3/2. To the negative 3/2. Times w to the fifth

Tohle je to samé jako 4 na (-3 lomeno 2) krát w na 5 na (-3 lomeno 2).

to the negative 3/2.

And once again this is just straight out of our exponent properties.

Znovu připomínám, tohle jsou vlastnosti mocnin.

Now, four to the negative 3/2,

4 na (-3 lomeno 2), přemýšlejme o tom.

let's just think about that. Four to the negative 3/2.

4 na (-3 lomeno 2).

We're using a different color just as a bit of an aside.

Použiji jinou barvu, na mezivýpočet.

So, four to the negative 3/2 is equal to,

4 na (-3 lomeno 2) je rovno…

that's the same thing as one over four to the 3/2,

Je to to samé jako (1 lomeno 4) na (3 lomeno 2).

and, let's see.

The square root of four is two and then we raise that to the third power, it's gonna be eight.

Odmocnina ze 4 je 2, to umocníme na 3, bude to 8.

So, this is equal to 1/8.

Je to tedy rovno (1 lomeno 8).

So, that's equal to 1/8, and so all of this

To se rovná (1 lomeno 8),

is going to be equal to 1/8.

tedy i tohle se rovná (1 lomeno 8).

And then w to the fifth, and then that

Pak w na 5, to celé na (-3 lomeno 2),

to the negative 3/2, we can multiply these exponents.

můžeme vynásobit exponenty.

That's going to be w to the five times negative 3/2.

To bude rovno w na [5 krát (-3 lomeno 2)].

Well, it's gonna be the negative 15/2 power.

Bude to w na (-15 lomeno 2).

So, I don't know which one you would say is simpler. This one over here or this one over here

Nevím, co je pro vás více zjednodušené, jestli tento zde nebo tento vedle,

but they are equivalent and they both are a lot simpler

ale oba výrazy jsou si rovny a oba jsou jednodušší,

than where we started.

než byly na začátku.

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