Simplifying $\sqrt[3]{\frac{16}{54}}$

Question

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

Answer 1

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}$

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

Answer 2

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.

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