Question

More squareroot problems.. Alg II

(picture)

Can you walk me through it? An answer is nice, but I want to understand it too. Thanks! :)

Answer 1

The list of perfect cubes are...

1, 8, 27, 64, ...

since 8 = 2^3, 27 = 3^3, etc etc

So can you factor 54 into two numbers such that one number is one of those perfect cubes?

Answer 2

Well 27 goes into 54 twice, so would it be 27 & 27?

Answer 3

close, but not quite

Answer 4

factor means write the number as a product of two (or more) numbers

so

54 = (some number) TIMES (some other number)

for example

Answer 5

Oh so it would be 27 & 2 because 27 * 2 = 54, right?

Answer 6

you got it

Answer 7

so

\[\Large \sqrt[3]{54}=\sqrt[3]{27*2}\]

\[\Large \sqrt[3]{54}=\sqrt[3]{27}*\sqrt[3]{2}\]

\[\Large \sqrt[3]{54}=\sqrt[3]{3^3}*\sqrt[3]{2}\]

\[\Large \sqrt[3]{54}=3\sqrt[3]{2}\]

Answer 8

Okay, that makes sense.

Answer 9

which means

\[\Large \frac{\sqrt[3]{54x^5}}{\sqrt[3]{2x^2}}\]

turns into

\[\Large \frac{3\sqrt[3]{2x^5}}{\sqrt[3]{2x^2}}\]

Answer 10

Can you factor \(\Large x^5\) at all (where one factor is a perfect cube)?

Answer 11

well x goes in 5 times so on the outside of the radical would it be x and inside the radical (x^2)?

Answer 12

so you're saying \[\Large \sqrt[3]{x^5} = x\sqrt[3]{x^2}\] ??

Answer 13

Yes, but I think its wrong...

Answer 14

that is correct

Answer 15

why do you think it is wrong?

Answer 16

so

\[\Large \frac{3\sqrt[3]{2x^5}}{\sqrt[3]{2x^2}}\]

becomes

\[\Large \frac{3x\sqrt[3]{2x^2}}{\sqrt[3]{2x^2}}\]

Answer 17

@panlac01 Well I thought it was wrong because I thought you may write x^3 on the outside of the radical, but you didn't write 3^3 so I wasn't sure... & @jim_thompson5910 Wouldn't that mean that the radical things cancel out leaving just 3x?

Answer 18

that's absolutely right, 3x is the final answer

Answer 19

provide some reading materials on radicals and exponents

Answer 20

it is correct because..

\[\Large \sqrt[3]{x^5} = \sqrt[3]{x^3*x^2}\]

\[\Large \sqrt[3]{x^5} = \sqrt[3]{x^3}*\sqrt[3]{x^2}\]

\[\Large \sqrt[3]{x^5} = x\sqrt[3]{x^2}\]

Answer 21

Sweet dandy bananas. Thanks guys so much! (:

Answer 22

you're welcome