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`