<PRE><B>Question 40419</B>
&lt;pre&gt;&lt;font size = 4&gt;&lt;b&gt;Solve this:
(9x&lt;sup&gt;6&lt;/sup&gt;y&lt;sup&gt;8&lt;/sup&gt;)&lt;sup&gt;1/6&lt;/sup&gt;

And I have this possible answers

a. xy(9y&lt;sup&gt;8&lt;/sup&gt;)&lt;sup&gt;1/6&lt;/sup&gt;
b. xy(3y)&lt;sup&gt;1/3&lt;/sup&gt; &lt;i&gt;&lt;font color = &quot;red&quot;&gt;&lt;------- that&#39;s the answer!&lt;/font&gt;&lt;/i&gt;
c. 3xy
d. x(9y&lt;sup&gt;2&lt;/sup&gt;)&lt;sup&gt;1/6&lt;/sup&gt;

========================================================

(9x&lt;sup&gt;6&lt;/sup&gt;y&lt;sup&gt;8&lt;/sup&gt;)&lt;sup&gt;1/6&lt;/sup&gt;

Write the 9 as 3&lt;sup&gt;2&lt;/sup&gt; 

(3&lt;sup&gt;2&lt;/sup&gt;x&lt;sup&gt;6&lt;/sup&gt;y&lt;sup&gt;8&lt;/sup&gt;)&lt;sup&gt;1/6&lt;/sup&gt;


Remove the parentheses by multiplying each inner exponent by the
outer exponent 1/6

3&lt;sup&gt;(2)(1/6)&lt;/sup&gt;x&lt;sup&gt;(6)(1/6)&lt;/sup&gt;y&lt;sup&gt;(8)(1/6)&lt;/sup&gt;

3&lt;sup&gt;2/6&lt;/sup&gt;x&lt;sup&gt;6/6&lt;/sup&gt;y&lt;sup&gt;8/6&lt;/sup&gt;

Reduce the fraction exponents

3&lt;sup&gt;1/3&lt;/sup&gt;x&lt;sup&gt;1&lt;/sup&gt;y&lt;sup&gt;4/3&lt;/sup&gt;

Change improper fraction exponent 4/3 to 1 1/3 but write it as 1+1/3

3&lt;sup&gt;1/3&lt;/sup&gt;x&lt;sup&gt;1&lt;/sup&gt;y&lt;sup&gt;1+1/3&lt;/sup&gt;

Now since we add exponents of like bases to multiply, we can use that 
fact in reverse, and when we see an exponent consisting of an addition,
such as in y&lt;sup&gt;1+1/3&lt;/sup&gt; we can write that as the multiplication y&lt;sup&gt;1&lt;/sup&gt;y&lt;sup&gt;1/3&lt;/sup&gt;.

3&lt;sup&gt;1/3&lt;/sup&gt;x&lt;sup&gt;1&lt;/sup&gt;y&lt;sup&gt;1&lt;/sup&gt;y&lt;sup&gt;1/3&lt;/sup&gt;

Now we put the two factors which have the same exponent 1/3 together

3&lt;sup&gt;1/3&lt;/sup&gt;y&lt;sup&gt;1/3&lt;/sup&gt;x&lt;sup&gt;1&lt;/sup&gt;y&lt;sup&gt;1&lt;/sup&gt;

Now since to remove parentheses we multiply the inner exponents by the
outer exponents, we can use that fact in reverse to say that when two
factors have the same exponents, we can put the product of the bases
inside parentheses and use the common exponent outside the parentheses,
so 3&lt;sup&gt;1/3&lt;/sup&gt;y&lt;sup&gt;1/3&lt;/sup&gt; can be written as (3y)&lt;sup&gt;1/3&lt;/sup&gt;

So the above becomes

(3y)&lt;sup&gt;1/3&lt;/sup&gt;x&lt;sup&gt;1&lt;/sup&gt;y&lt;sup&gt;1&lt;/sup&gt;

We can erase the 1 exponents and get

(3y)&lt;sup&gt;1/3&lt;/sup&gt;xy

which is the same as

xy(3y)&lt;sup&gt;1/3&lt;/sup&gt;

which is answer b.

Edwin
AnlytcPhil@aol.com&lt;/pre&gt;



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