# The Degree with the Irrational Exponent

Let′s suppose a is irrational number. What is the sense of record a^a , where a - positive number? Let′s consider three events: a=1 , a>1 , 0<a<1 .

1. If a=0 , then we suppose 1^a=1 .
2. Suppose a>1. Let′s take any rational number r_1<a and any rational number r_2>a. Thereat r_1<r_2 and a^(r_1)<a^(r_2) . In this case a^a means the number, that is between a^(r_1) and a^(r_2) for any rational numbers r_1 and r_2 such, that r_1<a and r_2>a. It is proved, that there is such number and singular for any a>1 and for any irrational number a.
3. Suppose 0<a<1. Let′s take any rational number r_1<a and any rational number r_2>a. Thereat r_1<r_2 and a^(r_1)>a^(r_2) . In this case a^a means such number, that is between a^(r_2) and a^(r_1) for any rational numbers r_1 and r_2, satisfying the inequality r_1<a<r_2. It is proved, that there is such number and singular for any number a with the interval (0,1) and any irrational number a.