What is a rational exponent?

Question

What is a rational exponent? 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, and simplify the expression, if possible.

Answer 1

     The rational numbers are anything that can be expressed as a fraction. It got its name stemming from the word ratio. So fractional exponents and rational exponents are the same thing.

      The numerator of the fractional exponent refers to a normal power, but the denominator refers to a root.

 

x^(2/3) means that whatever the value of x, you need to break up that fraction into 2 separate steps and square the number, then take the cube root of the number. You can also do it by taking the cube root first, and then squaring that. The second way is generally preferred because you deal with smaller numbers.

 

So 8^(2/3) could be done as 8 squared or 64, followed by the cube root of 64,which you may know is 4. Using the method

                   where you do the root first, the cube root of 8 is 2, and then 2 squared is 4. Ithink you'll admit that the second method 

                   easier.

So 125^(2/3) is125 squared then cube rooted. This would result in figuring out the cube root of 15625.......................=25

or do the cube root of 125, then square it. The cube root of 125 = 5 which is then squared resulting in 25.....easier method.

 

So 36^(3/2) . Do you want to figure out 36 cubed, then sqrt that? Not me. the sqrt 36 is 6 and 6 cubed is 216.

You should memoriz the perfect cubes just like you did when you first studied perfect squares.

1^3=1 2^3=8 3^3=27 4^3=64 5^3=125 6^3=216 . You can memorize more but this usually covers what you need.

Answer 2

A rational exponent is an exponent expressed as a ratio or fraction, such as x^1/2 or x^-2/3.  The denominator of the fraction represents a root:

 

x^1/2 = √x

 

x^1/3 = ³√x     where ³√ is the cube root

 

x^2/3 = ³√(x²)    (the cube root of x squared)

 

In general:

 

x^a/b = ^b√(x^a)

 

Example:

 

(x-1)^3/2 = √(x-1)³ = √{(x-1)²(x-1)} = (x-1)√(x-1)