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{{ printedBook.courseTrack.name }} {{ printedBook.name }} For any real number $a,$ the radical expression $na_{n} $ can be simplified as follows. $na_{n} ={∣a∣ifnis odd∣a∣ifnis even $ Since the radical is a real number and the index of the root is even, the expression underneath the radical is positive. Otherwise, the radical would be imaginary. $64ab_{4}c_{5} $ With this in mind, let's consider the possible values of the variables, $a,$ $b,$ and $c.$

- In the radical, the index is
**even**and the exponent of $b$ is**even**. Therefore, the expression will be real whether the value of $b$ is positive, negative or equal to $0.$ - In the radical, the index is
**even**and the exponents of $a$ and $c$ are**odd**. Since $b_{4}$ is always positive, in order for this radical expression to result in a real number, the product of $a$ and $c_{5}$ must be also positive — $a$ and $c_{5}$ must have the same sign.

$64ab_{4}c_{5} $

WritePowWrite as a power

$8_{2}ab_{4}c_{5} $

WriteSumWrite as a sum

$8_{2}ab_{4}c_{1+4} $

SumInExponentOne$a_{1+m}=a⋅a_{m}$

$8_{2}ab_{4}cc_{4} $

SplitIntoFactorsSplit into factors

$8_{2}ab_{2⋅2}cc_{2⋅2} $

ProdInExponent$a_{m⋅n}=(a_{m})_{n}$

$8_{2}a(b_{2})_{2}c(c_{2})_{2} $

CommutativePropMultCommutative Property of Multiplication

$8_{2}(b_{2})_{2}(c_{2})_{2}ac $

ProdPowII$a_{m}b_{m}=(ab)_{m}$

$(8b_{2}c_{2})_{2}ac $

RootProd$a⋅b =a ⋅b $

$(8b_{2}c_{2})_{2} ac $

SqrtToAbs$a_{2} =∣a∣$

$∣∣∣ 8b_{2}c_{2}∣∣∣ ac $