Question

"What is the ninth root of 32^3 in simplified radical form with the smallest possible index?"
A. 2 * square root of 4
B. 2 * third root of 2
C. 2 * third root of 4
D. 4 * third root of 3

Answer 1

We can write the 9th root like this, as a rational expression.
\[\large \left(32^3\right)^{1/9}\]
Using our laws of exponents, we multiply the powers,\[\large 32^{3/9}\]Which simplifies to,\[\large 32^{1/3}\]

To simplify further we want to find a factor of 32 that is a `perfect cube`.

Answer 2

\[\large 32 = 8\cdot 4\]Hmm it has a factor of 8.
\[\large 8=2\cdot2\cdot2=2^3\]
8 is a perfect cube!
We can take the cube root of 8.\[\large (32)^{1/3} \qquad = \qquad (8\cdot4)^{1/3} \qquad = \qquad 2(4)^{1/3}\]Understand how that works? :)

Answer 3

Thank you so much! (: