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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.

$ntimesna ⋅na ⋅…⋅na =aor(na )_{n}=a$

The $n_{th}$ root can also be expressed using a rational exponent.

$na_{1} =a_{n1}$

When a number is raised to the power of a fraction, that fraction is the number's **rational exponent**. Such an expression is equivalent to a root.

$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

To begin, notice that the expression $5x $ is written in radical form. Thus, it needs to be rewritten with a rational exponent. Recall that $na_{m} =a_{nm}.$ Notice that in $5x ,$ $x$ does not have an exponent. That means it is raised to the power of $1.$ This gives $5x =x_{51}.$

This expression has a rational exponent. Thus, we must rewrite it with a radical. Recall that $a_{nm}=na_{m} .$ Since the denominator of the exponent is $3,$ we can write the cube root of $x_{2}.$ $x_{32}=3x_{2} $

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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