Definition of a Rational Exponent.
a m/n = ( n √ a ) m
In general, if is a real number, then
a 1/ n =
The denominator of a fractional exponent is equal to the index of the radical.
So 8 1/3 is the exponential form of the cube root of 8 , and is its radical form .
Next, we ask: what sense can we make of a symbol like a 2/3 ? According to the rules of exponents:
a 2/3 = ( a 1/3 ) 2 or a 2/3 = ( a 2 ) 1/3
Use can use either of these rules to simplify the expression 8 2/3 as shown below.
8 2/3 = (8 1/3 ) 2 = 2 2 = 4
8 2/3 = (8 2 ) 1/3 = 64 1/3 = 4
Notice that we get the same answer either way. However, to evaluate a fractional power, it is more efficient to take the root first.
When you see a radical expression,
The denominator of a fractional exponent indicates the root.
Specifically, = a 1/3 or in general: = a m / n
Notice: The index of the radical becomes the denominator of the rational power, and the exponent of the radicand (expression inside the radical) becomes the numerator.
0 out of 0 correct.