#
Section 8-3

#
Rational Exponents

##
POWERS OF 1/2

You know from the
properties of exponents
that:

**
***
a
*
^{
b
}
·
*
a
*
^{
c
}
=
*
a
*
^{
b
+
c
}

You can use this to show that 9
^{
1/2
}
= ±3:

**
9
**^{
1/2
}
· 9
^{
1/2
}
= 9
^{
1/2 + 1/2
}
= 9
^{
1
}

So,

**
9
**^{
1/2
}
· 9
^{
1/2
}
**
= 9
**

What number multiplied by itself equals 9? There are two answers: 3 and –3.

**
3
**^{
}
· 3
^{
}
**
= 9
**

**
(
**
–
**
3)
**^{
}
· (
–
**
3)
**^{
}
**
= 9
**

So, raising a number to the power of 1/2 works almost the same as square roots (except that, with square roots, by convention, we usually only mean the positive answer).

##
OTHER FRACTIONAL POWERS

You can use the same property of exponents to show that 8
^{
1/3
}
= 2:

**
8
**^{
1/3
}
· 8
^{
1/3
}
· 8
^{
1/3
}
= 8
^{
1/3 + 1/3 + 1/3
}
= 8
^{
1
}

So,

**
8
**^{
1/3
}
· 8
^{
1/3
}
· 8
^{
1/3
}
**
= 8
**

What number multiplied by itself three times equals 8? The only answer in this case is 2.

**
2
**^{
}
· 2
^{
}
· 2
^{
}
**
= 8
**

By a similar logic, 8
^{
2/3
}
= 4, 8
^{
4/3
}
= 16, 8
^{
5/3
}
= 32, etc. (Convince yourself of this!)

##
CUBE ROOTS AND OTHER RADICALS

Fractional exponents can also be written as radicals:

The only difference is that,
**
in most books
**
, a radical expression is always positive, whereas the rational exponent may yield a positive or negative answer.

It's a good idea to review the properties of exponents before working through the problems in this section.