# simplify the cube root of a fraction

I am trying to solve a problem for algebr and keep getting a different answer then my text book. The problem involves cube roots. I am wondering how I am getting this wrong and how to solve it.

Problem : $$\sqrt[3]{\frac{16}{54}}$$

My answer : $$\frac{2 \sqrt[3] 2}{ 3\sqrt[3] 2}$$

Book : $$\sqrt[2]{\frac{8}{27}}$$

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@DonAntonio using \dfrac makes it unable to be read the cube root.OP please change it back –  ABC Apr 9 '13 at 2:36
@exploringnet, I don't understand what you mean: in my page all the roots appeared fine. Now I changed to double dollar signs to center them and make them bigger... –  DonAntonio Apr 9 '13 at 2:39
@DonAntonio Nope the $3$ was so small that it gets mixed up in itself. Now it's allright though,. –  ABC Apr 9 '13 at 2:41
@exploringnet, believe me: the odds your sight is worse than mine aren't good. Perhaps you've defined too small your page fonts in your browser? Holding the Ctrl. button and rolling the mouse's little wheel up or down you can increase/decrease the size of your pages...the little $\,3$'s really appeared clear to me. All of them. –  DonAntonio Apr 9 '13 at 2:45
Haha! Well it was right my page Zoom was 85% and now changed it to 110% .It's now right. –  ABC Apr 9 '13 at 2:49

Both are one and the same thing!

### You're Right!

$\dfrac{2 \sqrt[3] (2)}{ 3\sqrt[3] (2)}$=$\dfrac 23$=$\sqrt[3]{8/27}$

$2^3=8,3^3=27$

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how is ˆ1/2 -> ˆ1/3 –  BDGapps Apr 9 '13 at 2:36
@BDGapps I'm sorry, i din't see that . it must be a misprint in the book. You are all right. –  ABC Apr 9 '13 at 2:38
ok great thanks –  BDGapps Apr 9 '13 at 2:39
My pleasure. It's frustrating if you get a wrong answer after a confident solving. –  ABC Apr 9 '13 at 2:39

There must be a typo...in the book's solutions, if they show $\sqrt[\Large 2]{\dfrac 8{27}}$ That happens all too frequently.

You are correct, save for the fact that you could have simplified by canceling the common term in the numerator and the denominator, as I did to move from the second to the final equality.

$$\left(\dfrac{16}{54}\right)^{1/3} = \left(\dfrac{2\cdot 8}{2\cdot 27}\right)^{1/3}= \sqrt[\Large 3]{\dfrac{8}{27}} = \dfrac 23$$

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how is ˆ1/2 -> ˆ1/3. The book has square root of 8/27 not cubed –  BDGapps Apr 9 '13 at 2:37
The book is "wrong": misprint. You were perfectly on the right track. ;-) –  amWhy Apr 9 '13 at 2:49