# The Properties of Powers with the Rational Exponents

For any number a is determined the operation of natural exponentiation; for any number a!=0 is determined the operation of raising to the zero and integer negative power; for any number a>=0 is determined the operation of raising to the positive fractional power; for any number a>0 is determined the operation of raising to the negative fractional power.

Example. Determine the following:

(6.25)^0.5*(1/16)^0.25-(-4)^(-1)*(0.343)^0 .

We have (6.25)^0.5=(25/4)^(1/2)=sqrt(25/4)=5/2 ; (1/16)^0.25=(1/16)^(1/4)=root(4)(1/16)=1/2 ; (-4)^-1=-1/4 ; (0.343)^0=1. As a result we obtain 5/2*1/2-(-1/4)*1=3/2 .

If a>0 , b>0 and r, s are any rational numbers, then:

1. a^r*a^s=a^(r+s)

2. a^r-:a^s=a^(r-s)

3. (a^r)^s=a^(rs)

4. a^r*b^r=(ab)^r

5. a^r/b^r=(a/b)^r