SOLUTION: Use rational exponents to write as a single radical expression. Assume that all variables represent positive real numbers. The 5 root of 3 multiplied by the cube root of z.

Algebra ->  Radicals -> SOLUTION: Use rational exponents to write as a single radical expression. Assume that all variables represent positive real numbers. The 5 root of 3 multiplied by the cube root of z.      Log On


   

Question 629380: Use rational exponents to write as a single radical expression. Assume that all variables represent positive real numbers.
The 5 root of 3 multiplied by the cube root of z.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
root%285%2C+3%29%2Aroot%283%2C+z%29
Radicals can be multiplied only if they are the same type of roots. Here we have two types, a 5th root and a cube (3rd) root. So we cannot multiply them together as they are now.

As the problem suggest, rational (fractional) exponents can be used to convert these roots so that we can multiply them. We start by rewriting the radical expressions with fractional exponents. The exponent for 5th roots is 1/5 and the exponent for cube roots is 1/3. So we can rewrite the above as:
3%5E%28%281%2F5%29%29%2Az%5E%28%281%2F3%29%29
Keeping in mind that the denominator represents the type of root, we can match the types of these roots by finding the lowest common denominator (LCD). The LCD of 3 and 5 is 15. Rewriting our expression with exponents having denominators of 15:
3%5E%28%283%2F15%29%29%2Az%5E%28%285%2F15%29%29
Using the fact that a%5E%28%28p%2Fq%29%29+=+root%28q%2C+a%5Ep%29, we can switch back to radical notation:
root%2815%2C+3%5E3%29%2Aroot%2815%2C+z%5E5%29
Since we have the same types of roots we can multiply these:
root%2815%2C+3%5E3%2Az%5E5%29
And finally we can simplify the 3%5E3:
root%2815%2C+27%2Az%5E5%29