Question

Difficult problem. I couldn't figure it out -- write in simplest radical form

Answer 1

\[(\frac{ 25 }{ \sqrt[3]{20} })^{\frac{ 2 }{ 3 }}\]

Answer 2

any ideas on how to start?

Answer 3

Haha that's where I zoned out... do I have to bring the \[\sqrt[3]{20}\] to the top?

Answer 4

no, distribute that 2/3 to start

Answer 5

but first, change the cubed root to a fraction

Answer 6

that would be \[20^{1/3}\]

Answer 7

then?

Answer 8

\[\frac{ 25^{2/3} }{ 20^{3/3} }\]

Answer 9

stop, 20^(3/3)? check that

Answer 10

Originally it was 20^1/3, I'm distributing the 2/3 power so shouldn't I add since it's the same base of 3

Answer 11

do you add or multiply exponents?

Answer 12

\[ \Large
(a^b)^c = a^{bc}
\]

Answer 13

\[ \Large
a^b \times a^c = a^{b+c}
\]

Answer 14

stahhpppp ranga

Answer 15

haha so wait, that was correct?

Answer 16

that's not necessary,mighty lion's knows that just made a mistake

Answer 17

no mighty, your answer was not there yet. You have \[\frac{25^{2/3}}{[20^{1/3}]^{2/3}}\]

Answer 18

correct? do you see?

Answer 19

Ok I see. So now I raise 20^1/3 to the 2/3 power... geez this is complicated

Answer 20

meh, not too, just apply the rules

Answer 21

\[25^{\frac{ 2 }{ 3 }} \over 20^{\frac{ 2 }{ 9 }}\]

Answer 22

good

Answer 23

so now, change it back into roots

Answer 24

Couldn't I just divide and get \[25^{\frac{ 4 }{ 9 }}\]

Answer 25

Oh, I have to get it to roots because of simple radical form?

Answer 26

no you can't divide

Answer 27

no, because it makes it clearer for the next step

Answer 28

So it would be \[\sqrt[3]{25^{2}} \over \sqrt[9]{20^{2}}\]

Answer 29

good so now, compute 25^2 then find the cubed root if you can

Answer 30

The cube root of 625 is \[5\sqrt[3]{5}\]

Answer 31

Is \[5\sqrt[3]{5 } \over \sqrt[9]{20^2}\] the final answer?

Answer 32

@jim_thompson5910

Answer 33

i bno, you have to simplify the bottom too

Answer 34

\[\sqrt[9]{400}\]

Answer 35

yea, now, can that be simplified any further?

Answer 36

I don't think so.

Answer 37

right, so we now want to try and make the top have a ninth root too

Answer 38

so that we can actually do something interesting

Answer 39

I don't know how to do that, never done a problem like this before

Answer 40

the simplification you started with may have been too much lets go back to the \(^{3}\sqrt {625}\)

Answer 41

so now for example, if I had 2 and I wanted it to be a cubed root, I would do this. \[^3\sqrt{2^3}\] you see how they undo each other and just leave you with two?

Answer 42

so, same idea here, you want a ninth root

Answer 43

but you have a 1/3

Answer 44

Ah yes I see. So you add a number to both sides.

So \[\sqrt[9]{625^6}\]

Answer 45

do you follow?

Answer 46

9/3=?

Answer 47

3

Answer 48

so it's \(625^?\)

Answer 49

\[625^{3}\] right?

Answer 50

yup, so now, remember how we distributed the 2/3 to start? we can now un-distribute(factor) the ninth root

Answer 51

so \[625^{\frac{ 3 }{ 9 }}\]

Answer 52

yea and now the whole fraction becomes?

Answer 53

(once you get this I'l show you a faster way, I went really really roundabout)

Answer 54

\[625^{\frac{ 3 }{ 9 }} \over \]\[\sqrt[9]{400}\]

Answer 55

I can't take all night to figure this out though, I have like 30 problems after this

Answer 56

yea, other way is faster jus undistribute the ninth root now then we can do the otehr way in 5 min

Answer 57

it's the same principle which is what is important

Answer 58

so what I just wrote is the correct answer?

Answer 59

no,not yet

Answer 60

Just restart, it takes like 2 min
\[(\frac{ 25 }{ \sqrt[3]{20} })^{\frac{ 2 }{ 3 }}\]

From the start, remember my example with 2 and cubed root of 2 cubed? Try and apply that to start

Answer 61

(to the 25)

Answer 62

\[\sqrt[3]{25^2}\]

Answer 63

squared or cubed on the 25?

Answer 64

Oh yeah, the other way around.

Answer 65

so write the whole problem out