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Radical expressions involve taking the root of a quantity. These expressions can be expressed with fraction exponents, or *rational* exponents. Since radicals and rational exponents are two different ways to write the same thing, translating between the two is useful.

The $n_{th}$ root of a number $a$ expresses another number that, when multiplied by itself $n$ times, will result in $a.$ Aside from the radical symbol, the notation is made up of the radicand $a$ and the index $n.$

The resulting number is commonly called a radical. For example, the radical expression $416 $ is the fourth root

of $16.$ $416 $ simplifies to $2$ because $2$ multiplied by itself $4$ times equals $16.$
$416 =42_{4} =2 $
The general expression $na $ represents a number which equals $a$ when multiplied by itself $n$ times.

$ntimesna ⋅na ⋅…⋅na =aor(na )_{n}=a$

Just as with exponents, the most common roots have special names: square roots and cube roots have an index of $2$ and $3,$ respectively.

$a_{nm}=na_{m} $

$58_{2} $

RootToPowD

$na =a_{n1}$

$(8_{2})_{51}$

PowPow

$(a_{m})_{n}=a_{m⋅n}$

$8_{2⋅51}$

MoveLeftFacToNumOne

$a⋅b1 =ba $

$8_{52}$

Rewrite the given expressions in the opposite form. $5x andx_{32}$

Show Solution

Simplify the following radical expression by rewriting it so that it has a rational exponent. $115_{33} $

Show Solution

Recall the following rule.
$na_{m} =a_{nm}$
Using this, it is possible to rewrite a radical expression so that it has a rational exponent. We will first use this rule and then simplify the resulting expression.
Thus, the expression equals $125.$

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