Radicals, expressed by the notation `\rootn(b)`, are also used to denote roots. The number `b` is the radicand, and the positive integer `n` is the index of the radicand.

Definition of `\rootn(b)`

If `n` is a positive integer and `b` is a real number such that `b^(1//n)` is a real number, then `\rootn(b)=b^(1//n)`

If the index `n` equals `2`, then the radical `\root2(b)` is written as simply `\sqrt(b)`, and it is referred to as the principal square root of `b` or simply the square root `b`. The symbol `\sqrt(b)` is reserved to represent the nonnegative square root of `b`. To represent the negative square root of `b`, write `-\sqrt(b)`. For example `\sqrt(25)=5`, whereas `-\sqrt(25)=-5`.

Definition of `(\rootn(b))^m`

For all positive integers `n`, all integers `m`, and all real numbers `b` such that `\rootn(b)` is a real number, then `(\rootn(b))^m=\rootn(b^m)=b^(m//n)`

The equations `b^(m//n)=\rootn(b^m)` and `b^(m//n)=(\rootn(b))^m` can also be used to write radical expressions in exponential form. For example,

`sqrt((2ab)^3)=(2ab)^(3//2)`

The definition of `(\rootn(b))^m` can be used to evaluate radical expressions. For instance,

`(\root3(8))^4=8^(4//3)=(8^(1//3))^4=2^4=16`

Definition of `\rootn(b^n)`

If `n` is an even natural number and `b` is a real number, then `\rootn(b^n)=|b|`

If `n` is an odd natural number and `b` is a real number, then `\rootn(b^n)=b`

Here are some examples of these properties.

`\root4(16z^4)=2|z| \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \root5(32a^5)=2a`

Properties of Radicals

If `m` and `n` are natural numbers and `a` and `b` are nonnegative real numbers then

Product `\ \ \ \ \ \ \ \ \ \ \rootn(a)*\rootn(b)=\rootn(ab)`

Quotient `\ \ \ \ \ \ \ \ \ \ \rootn(a)/\rootn(b)=\rootn(a/b)`

Index `\ \ \ \ \ \ \ \ \ \ \rootm(\rootn(a))=\root(mn)(a)`

A radical is in simplest form if it meets all of the following criteria.

1. The radicand contains only powers less than the index. (`\sqrt(x^5)` does not satisfy this requirement because `5`, the exponent, is greater than `2`, the index.)
2. The index of the radical is as small as possible. (`\root9(x^3)` does not satisfy this requirement because `\root9(x^3)=x^(3//9)=x^(1//3)=\root3(x)`.)
3. The denominator has been rationalized. That is, no radical appear in the denominator. (`1/sqrt(2)` does not satisfy this requirement.)
4. No fractions appear under the radical sign. (`\root4(2//x^3)` does not satisfy this requirement.)

Like radicals have the same radicand and the same index. For instance,

`3\root3(5xy^2) \ \ \ \ \ ` and ` \ \ \ \ \ -4\root3(5xy^2)`
are like radicals. Addition and subtraction of like radicals are accompanied by using the distributive property. For example,
`4sqrt(3x)-9sqrt(3x)=(4-9)sqrt(3x)=-5sqrt(3x)`
`2\root3(y^2)+4\root3(y^2)-\root3(y^2)=(2+4-1)\root3(y^2)=5\root3(y^2)`
The sum `2sqrt(3)+6sqrt(5)` cannot be simplied further because the radicands are not the same. The sum `3\root3(x)+5\root4(x)` cannot be simplified because the indices are not the same.

Sometimes it is possible to simplify radical expressions that do not appear to be like radicals by simplifying each radical expression.

Multiplication of radical expressions is accomplished by using the distributive property. For instance,

`sqrt(5)(sqrt(20)-3sqrt(15))=sqrt(5)(sqrt(20))-sqrt(5)(3sqrt(15))`
` \ \ \ \ \ = sqrt(100)-3sqrt(75)`
` \ \ \ \ \ = 10-3*5sqrt(3)`
` \ \ \ \ \ = 10-15sqrt(3)`

The product of more complicated radical expressions may require repeated use of the distribution property.

To rationalize the denominator of a fraction means to write it in an equivalent form that does not involve any radicals in its denominator.

To rationalize the denominator of a fractional expression such as

`1/(sqrt(m)+sqrt(n))`
we make use of the conjugate of `sqrt(m)+sqrt(n)`, which is `sqrt(m)-sqrt(n)`. The product of these conjugates does not involve a radical.

`(sqrt(m)+sqrt(n))(sqrt(m)-sqrt(n))=m-n`