Question

How do you simplify (3rd root of 5) divided by (sqrt of 5)?

Answer 1

`root(6)(3125)/5`

Explanation:

You won't be able to multiply or divide them until they have the same index. So to simplify this, we will make them have the same index.
`root(3)(5)/sqrt5`

Using the Law of Indices, we can rewrite this as:
`=5^(1/3)/5^(1/2)`

As you can see, they both have fraction exponents now. We must now make those fraction exponents have the same denominator. You can use any multiple of 2 and 3, but the easiest to use is 6 (since that is the least common denominator).
`=5^[(1/3)(2/2)]/5^[(1/2)(3/3)]`
`=5^[2/6]/5^[3/6]`

Using the Law of Indices again, you can rewrite this as:
`=root(6)(5^2/5^3)`
`=color(red)root(6)(1/5)`

You can have that as your final answer. However, you can still rationalize this if you want. Simply multiply something that will remove the radical from the denominator.
`=root(6)(1/5)`
`=root(6)(1)/root(6)(5)`
`=root(6)(1)/root(6)(5)*root(6)(5^5)/root(6)(5^5)`
`=root(6)(1*5^5)/root(6)(5^6)`
`=root(6)(5^5)/5`
`=color(red)(root(6)(3125)/5)`