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P.4. Radicals and Rational Exponents

Thus far we have only discussed integer exponents. What about fractional (rational)
exponents? What would we mean by 2^{1/2}, for instance? In order to define
such expressions in a meaningful way, it is useful to first look at some useful
properties of exponents. We state these properties without proof. The symbols a, b,
and c stand for arbitrary nonzero real numbers with a positive, and b positive as well in (P.4.5).

**(P.4.1):**a^{b+c}= a^{b}a^{c}

**(P.4.2):**a^{-b}= 1/a^{b}

**(P.4.3):**a^{b-c}= a^{b}/a^{c}

**(P.4.4):**(a^{b})^{c}= a^{bc}

**(P.4.5):**a^{c}b^{c}= (ab)^{c}

Equation (P.4.4) is the one we will use to define rational exponents. To see this,
consider the case a = 2, b = 1/2, and c = 2. According to (P.4.4), we see that
(2^{1/2})^{2} = 2^{(1/2)*2} = 2^{1} = 2. Thus, 2^{1/2} is the
number whose square is equal to 2. In other words, 2^{1/2} is the * square root *of 2, more commonly written as
√2. More generally, for any positive
number a, a

^{1/2}is the square root of a, more commonly written as √a. Similarly, a

^{1/3}is the cube root of a, i.e. the number whose cube is equal to a, which is also written as

^{3}√a. To fully generalize, for every positive real number a and every positive integer k, a

^{1/k}is the kth root of a, i.e. the number whose kth power is equal to a. This is more commonly written as

^{k}√a. Such an expression is known as a

*and the symbol*

**radical expression**^{k}√ is known as a

*.*

**radical**

**Example 1:** Simplify 2^{1/2} and 4^{1/2}.

**Solution: **From the above discussion, we know that 2^{1/2}=√2, which cannot be further simplified. We also know that 4^{1/2}=√4=2 since 2^{2}=2*2=4.

**Example 2:** Simplify 3^{1/3} and 3^{2/3}^{}.

**Solution: **From the above discussion, we know that 3^{1/3} = ^{3}√3, which cannot be further simplified. We also know from (P.4.4) that

3^{2/3}=3^{2*1/3}=(3^{2})^{1/3}=9^{1/3}=^{3}√9.

**Example 3:** Simplify 1000^{1/3} and 1000^{2/3}^{}.

**Solution: **We have 1000^{1/3} = ^{3}√1000 = 10, since 10^{3} = 10*10*10 = 1000. We also know from (P.4.4) that

1000^{2/3}=1000^{1/3*2}=(1000^{1/3})^{2}=10^{2}=100.

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