Rational Exponents

Rational Exponents

Expressions like  a1/2 , 5-1/4 ,and (2y)4/5 have not yet been defined. We will define

such expressions so that the general properties of exponents hold.

Consider a1/2.a1/2. If we want to multiply by adding exponents, it must

follow that a1/2 . a1/2 = a1/2 +1/2, or a1 Thus we should define a1/2 to be a

square root of a. Similarly, a1/3.a1/3 a1/3 = a1/3+1/3+1/3,or  a1,  so a1/3

should be defined to mean \sqrt[3]{a}.

 

a1/n

For any nonnegative real number a and any natural number index n

(n ≠1)

a1/n  means  \sqrt[n]{a} (the nonnegative nth root of a).

am/n

For any natural numbers m and n (n ≠1)and any nonnegative real

number a,

am/n means  \sqrt[n]{a^{m}} , or (\sqrt[n]{a})^{m}

 

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