What is a rational exponent and how are rational exponents related to radicals? Give an example of how an expression with a rational exponent can be rewritten as a radical expression?  

Expert Answers

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You need to remember the relation that exists between a rational exponent and a radical, such that:

x^(m/n) = root(n)(x^m)

Hence,the radical symbol can be converted into a rational exponent, whose denominator represents the order of radical.

You may start with the following exponent operation, such that:

x^(m/n) = (x^(1/n))^m

You also can replace y for x^(1/n) such that:

y = x^(1/n)

Raising both sides to nth power, you can get eliminate the rational exponent, such that:

`y^n = (x^(1/n))^n => x = y^n`

Hence, evaluating the nth root of x yields x^(1/n).

You may consider the following example, such that:

`root(3)(16) = 16^(1/3)`

You may replace `4^2` or `2^4` for `16` , such that:

`root(3)(16) = root(3)(4^2) = 16^(1/3) =(4^2)^(1/3) = 4^(2/3)`

`root(3)(16) = root(3)(2^4) = (2^4)^(1/3) = 2^(4/3)`

You may notice that the order of radical (3) becomes the denominator of rational exponent and the numerator is the power(exponent) of the number under the radical.

Hence, you may convert the radical into a rational exponent, such that:`root(3)(16) = 4^(2/3) =2^(4/3)`

Hence, if you need to perform the conversion between radical in rational exponent or conversely, you need to use the formula `x^(m/n) = root(n)(x^m).`

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