Expert: Frederick Koh - 12/27/2011

Question
QUESTION: Topic : Rational Exponents

I need to write each expression in radical form, or write each radical in exponential form. Can you please show the process?

A) ³√5xy^2
B) ⁴√625x^2  (Correct answer according to book is: 5x^ 1/2. I got 5x^ 1/4, considering the "⁴" in ⁴√625x^2).

Also, can you please show me how to correctly simplify:

³√6x^2/³√5y

--Thanks

ANSWER: A) ³√5xy^2 = (5xy^2)^ (1/3)
         = 5^(1/3) x^(1/3) y ^[2*(1/3)]
         = 5^(1/3) x^(1/3) y ^(2/3)   (shown)

  (Note that we use this law (abc)^(1/n) =  a^(1/n)*b^(1/n)*c^(1/n) ;
   in the above question we have a=5, b=x and c=y^2)

B) ⁴√625x^2 = (625 x^2)^(1/4) = 625^ (1/4) x^[2*(1/4)]
         = 5x^(1/2)     (shown)


³√6x^2/³√5y = [(6 x^2 ) / (5y)] ^(1/3)
         = [6/5 * x^2 * y^(-1)]^ (1/3)    (Note that y is in the denominator, hence
         in order to bring it to the numerator, we rewrite
         the 1/y part as y^-1 )
         
         = (6/5)^(1/3) x^ (2/3) y^(-1/3)     (shown)

Hope this helps. Peace.

---------- FOLLOW-UP ----------

QUESTION: I'm sorry I'm posing an inconvenience with these petty questions, promise these will be the last for a while.

Thank you so much for the previous solutions, but I think there was a slight misunderstanding for the last part to my question.

Topic:  Using a conjugate to rationalize a denominator (no rational exponents here).

A) ³√6x^2/³√5y

The next few problems, on  the other hand, are a comeback to rational exponents.

A) Word Problem : A women's regulation-sized basketball is slightly smaller than a man's basketball.  The radius r of the ball that holds V cubic units of air is (3V/4π)^1/3. I need to find the radius of the women's basketball and men's basketball,each.

Given measurements:
Men's basketball= 455 in.^3.
Women's basketball= 413 in.^3.

These are my answers. Are they correct? If so, is further simplification necessary?

Men's:³√1365/4π
Women's:³√1239/4π

Simplify:
B)⁴√64 (My answer: 3)
C)⁶√216 (Answer according to book: √6. I don't understand).

--Thanks so much

Answer
A) ³√6x^2/³√5y = [³√6x^2*(³√5y )^2]/ [(³√5y) *(³√5y )^2]

    [multiply both numerator and denominator by (³√5y )^2 ]

         = [(6x^2)^(1/3) * (5y)^(2/3)]/(5y)
         
         = [(6x^2)^(1/3) * (25y^2)^(1/3)]/(5y)

         =  (150 x^2 y^2)^(1/3) / (5y)

         =  ³√(150 x^2 y^2)/(5y)    (shown)

A) Men's:³√1365/4π
  Women's:³√1239/4π

  Both answers are correct; there is no need for further simplification.

B)⁴√64  = (64)^(1/4)

       = (8^2)^(1/4)

       = 8^(2*1/4)
    
        =8^(1/2) = √8   (shown)

C) ⁶√216 = (216)^(1/6)
       
        =  (6^3)^(1/6)

        = 6 ^( 3*1/6)

        = 6 ^(1/2) = √6  (shown)

Hope this helps. Peace.