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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 thefourth rootof 16. $416 $ simplifies to 2 because 2 multiplied by itself 4 times equals 16.

$ntimesna ⋅na ⋅…⋅na =aor(na )_{n}=a$

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

$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.

Show Solution

Recall the following rule.
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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