Which is equivalent to rootindex 4 startroot 9 endroot superscript one-half x?

The above relations enable us to express radicals as fractional exponents and fractional exponents as radicals.

EXAMPLES 1. root(5,3)=3^(1/5)

2. root(3,2^2)=2^(2/3)

3. root(x+3)=(x+3)^(1/2)

4. x^(4/5)=root(5,x^4)

5. 3x^(3/4)=3root(4,x^3)

6. x^(1/2)y^(2/3)=root(x)root(3,y^2)

When the value of a radical expression is a rational number, we say it is a perfect root Since root(n,a^nk)=a^k, a radical expression is a perfect root if the radicand can be expressed as a product of factors each to an exponent that is an integral multiple of the radical index.

The value of the radical is obtained by forming the product of the factors. where the exponent of each factor is its original exponent divided by the radical index.

EXAMPLES 1. root(5^6)=5^(6/2)=5^3

2. root(x^10)=x^(10/2)=x^5

3. root(3,8x^6y^9=root(3,2^3x^6y^9=2^(3/3)x^(6/3)y^(9/3)=2x^2y^3

Note Nonperfect roots such as root(2),root(3,2),root(3),root(4,5),root(5,4),1+root(2) and 5-root(3,9) are irrational numbers. An irrational number is a number that cannot be expressed in the form p/q, where p,q∈{Iota},q!=0.

Note 1. Since a^(m/n)=a^(mk/nk) for all a>0, a {is-in} {real}, and m,n∈N, k {is-in} {rat}, k>0, we have root(n,m)=root(nk,a^mk), provided nk and mk∈N.

root(3,a)=root(6,a^2) and root(n,1)=1

2. 1^n=1 and root(n,1)=1.

10.2 Standard Form of Radicals

THEOREM If a,b∈R , a>0,b>0, and n∈N then root(n,ab)=root(n,a)root(n,b).

Proof root(n,ab)=(ab)^(1/n)=a^(1/n)b^(1/n)=root(n,a)root(n,b)

EXAMPLES 1. root(32)=root(2^5)=root(2^4*2^1)

=root(2^4)*root(2)

=2^2root(2)=4root(2)

2. root(16x^3y=root(2^4x^3y=root(2^4x^2xy)

=root(2^4x^2)root(xy)

=2^2xroot(xy)

=4xroot(xy)

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