Question

What is a simpler form of the radical expression?

3^sq root of 125x^21y^24

5x^7|y^8|
15x^21|y^24|
15|x^21|y^24
5x^7y^8

Answer 1

Do you mean \[3^{\sqrt{125x^{21}y^{24}}}\]

Answer 2

or this
\[\sqrt[3]{125x^{21}y^{24}}\]
?

Answer 3

The second one

Answer 4

does splitting it up like this:
\[\sqrt[3]{125}\sqrt[3]{x^{21}}\sqrt[3]{y^{24}}\]
help you out?

Also, \[\sqrt[b]{x^{a}} = x^{\frac{a}{b}}\]

Answer 5

Im a little lost.

Answer 6

(a*b*c)^2 = a^2 * b^2 * c^2

Answer 7

^
Is that the formula?

Answer 8

yes, you can use any number, not just 2

Answer 9

which is why I suggested \[\sqrt[b]{x^{a}}=x^{\frac{a}{b}}\]

Answer 10

so the whole thing becomes
\[(125x^{21}y^{24})^{\frac{1}{3}}\]

Answer 11

and then you can use N00bstyle's "formula" to solve

Answer 12

\[\sqrt[3]{125}=125^{1/3}=(5^{3})^{1/3}=5\]

Answer 13

This is one example, you can apply this to the other two...let me know if you manage to do that

Answer 14

so 125x^21=1953125?

Answer 15

@_@ im too confused

Answer 16

Okay first rule when you have multiplications under a square root:
\[\sqrt{ab}=\sqrt{a}\sqrt{b}\]

Answer 17

Do you understand this?

Answer 18

uhmm, yes

Answer 19

Okay, you have a multiplication of 125*x^21*y^24

Answer 20

oh, okay! Im getting it.

Answer 21

But how would multiplying make a simpler form?

Answer 22

\[\sqrt[3]{125}*\sqrt[3]{x^{21}}*\sqrt[3]{y^{24}}\]

Answer 23

NOW you apply the rule which aachem said:
\[\sqrt[y]{x}=x^{1/y}\]

Answer 24

?

Answer 25

To be more precise:
\[\sqrt[y]{a^{x}}=a^{x/y}\]

Answer 26

So when you rewrite \[\sqrt[3]{125}\]
You get:
\[125^{1/3}\]

Answer 27

If you don't understand, just tell me :)

Answer 28

I dont understand ):

Answer 29

so 125*.5=62.5

Answer 30

that's correct, but that's not what you want to know :)

Answer 31

what do i need to know?

Answer 32

D:

Answer 33

Read the first message aachem wrote you

Answer 34

Sorry, third message

Answer 35

OH! THE SQUARE ROOT!

Answer 36

He splits up all the multiplications

Answer 37

afterwards, you just need to apply the rule he gave you. Later in the text, I gave you an example about the cubic root of 125

Answer 38

and multiply's each number by the power of 3?

Answer 39

he takes the cubic root of every numbers yes

Answer 40

that's what the rule says:
\[\sqrt{ab}=\sqrt{a}\sqrt{b}\]

Answer 41

This also applies to:
\[\sqrt[3]{abc}=a^{1/3}b^{1/3}c^{1/3}\]

Answer 42

...

Answer 43

Let me write it for you, it will give you the answer, but it's important that you know what happens okay

Answer 44

Yes

Answer 45

To me it's important you know how the rules can be applied.
When you know the rules you can apply it to every equation...

Answer 46

k

Answer 47

\[\sqrt[3]{125x^{21}y^{24}}=\sqrt[3]{125^{1} x^{21}y^{24}}=\sqrt[3]{125}\sqrt[3]{x^{21}}\sqrt[3]{y^{24}}\]
This is the first part of the reformulation of the equation

Answer 48

So you split them

Answer 49

Then divide?

Answer 50

\[125^{1/3}(x^{21})^{1/3}(y^{24})^{1/3}\]

Answer 51

ok, you divide by 3

Answer 52

OH!

Answer 53

Now I applied the 'second' rule:
\[\sqrt[y]{a^{x}}=a^{x/y}\]

Answer 54

that's why there are all 1/3 in the equation

Answer 55

Its making sense now!

Answer 56

Now you will have to apply rules of multiplying exponents...do you know that?

Answer 57

OK, I will give it a try!
You must remember that to be able to easily calculate a third root, you'll need a number that has 3 same factors. E.g.:\[\sqrt[3]{125}=5\]because \[125=5^3\]You have to simplify:\[\sqrt[3]{125x^{21}y^{24}}=\sqrt[3]{5^3x^{21}y^{24}}\]Now, part of your problem is already done, the thrird root of 125 is 5, so we have left:\[5\sqrt[3]{x^{21}y^{24}}\]Oh, but now you can see that all these factors x can also be grouped in three same powers:\[x^{21}=x^7x^7x^7\]and \[y^{24}=y^8y^8y^8\]So we can calculate the third root of these numbers, just like we did for 125:\[\sqrt[3]{x^7x^7x^7}=x^7\]and\[\sqrt[3]{y^8y^8y^8}=y^8\]So the end result must be:\[\sqrt[3]{125x^{21}y^{24}}=5x^7y^8\]
The only thing you must remember is: make 3 equal groups of powers.

What if you had to do this:\[\sqrt[3]{x^{22}}\]Obviously, 22 is not a multiple of 3, but we will do as much as we can: \[\sqrt[3]{x^{22}}=\sqrt[3]{x^7x^7x^7x}=x^7\sqrt[3]{x}\]Everything goes as expected, only the last x is stuck under the radical. Bad luck!

Answer 58

so:
\[(x^{21})^{1/3}=x^{7}\]

Answer 59

and 1/3*125=41.666666666666666666666666666667

41.666666666666666666666666666667 squared is 6.4?

Answer 60

so if its 6.4 with x^7 it must be 5x^7y^8?

Answer 61

0_0

Answer 62

I don't know what you mean by 41.6666666666 squared is 6.4.
When you square something, you have 41.66666^2 = 41.66666 * 41.66666 = a big number

Answer 63

oops, i mean square root

Answer 64

ah okay, I see too

Answer 65

5x^7y^8 is idd the answer, but you will have to take a look at:
exponents rules and mutiplication with square roots

Answer 66

Defiantly

Answer 67

whew

Answer 68

Just try to remember: \[\sqrt[y]{a^{x}} = a^{x/y}\]
So for example: \[\sqrt[3]{8}=8^{1/3}=(2^{3})^{1/3}=2^{3*1/3}=2^{1}=2\]

Answer 69

Yes, il take notes just in case I forget, Thank you!

Answer 70

You'll get used to in time...just practise a lot ;)

Answer 71

Thanks for your patience! I can be allot to deal with!

Answer 72

Ha, you're welcome, good luck.