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

1 Answer
Rafael
Sep 21, 2015

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)

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