The n-th Roots(or Radicals)

Introduction Roots (or radicals) are the inverse of exponents (or powers). Examples 32=9 → =3 …() 53=125 → =5 24=16 → =2 () A radical without an index is understood to be a square root : = is called ”The Square Root”, is called ”The Cube Root”, … is called ”The n-th Root”.	Symbols
As is shown in the right figure, • if a>0 and n is an Even positive integer, xn=a → 2 real x values satisfy the equation. The positive value is denoted by . The negative value is denoted by −. • if a>0 and n is an Odd positive integer, xn=a → Only 1 real x value satisfies the equation. The positive value is denoted by . • if a>0 and n is an Even positive integer, xn=−a → No real x value satisfies the equation. • if a>0 and n is an Odd positive integer, xn=−a → 1 real x value satisfies the equation. The negative value is denoted by . • if a=0, either n is Even or Odd, xn=0 → x=0 =0 (Always 1 value.) In the following we will deal with positive radicands and those of positive n-th roots. Therefore, either n is even or odd, there exists only one positive value denoted by for each a, n. (See the blue curve of the right Figure.)	Even index and Odd index
Basic Rule to simplify the n-th radicals. To simplify a radical means to remove nth powers from the radicand (the number inside the radical). You can simplify a radical if the radicand has the powers greater than or equal to the index n. =a If b is not expressed by the n-th power of any integer, =a By definition xn=a → x= So When x=a (a>0) xn=an → x= ∴=a	Examples (1)==2 (2)==2 (A radical without an index is understood to be a square root.) (3)===2 (4)==5 (5)==2 (6)==2
QuestionSimplify the radicals. (1) = Check Erase Help =2×3=6 (2) = Check Erase Help ==4 (3) = Check Erase Help ==2	(4) = Check Erase Help Factor the radicand(the number inside the radical) =5 (5) = Check Erase Help Factor the radicand(the number inside the radical) =2 (6) = Check Erase Help Factor the radicand(the number inside the radical) ==2

Laws of Radicals

Suppose a>0, b>0 and m, n, p are positive integers.

(1)= …(1)

(2)= …(2)

(3)( )m= …(3)

(4)= …(4)

(5)= …(5)

Sorry to say, these rules are used only as a guide.
You had better calculate by the equivalent Fractional Exponents rather than by these Laws of Radicals directly.

(Proof)
(1)←

Let x=

Then xn=()n=ab

According to the definition : xn=a → x=

Therefore =

Similarly, (2) can be shown.
(3)←

Let x=()m

Then xn=()mn=(()n)m=am

According to the definition : xn=a → x=

Therefore ( )m=

Examples

(1)=

(2)=

(3)( )4=

(4)=

(5)=

(4)←

Let x=

Then xmn=()mn=(()m)n

As ()m=

xmn=()n=a

Let y=

Then ymn=()mn=a

Therefore x=y (x, y>0)

(5)←

Let x=

Then xnp=()np=amp

Let y=

Then ynp=()np=(()n)p=(am)p=amp

Therefore x=y (x, y>0)

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