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Simplify the expression.  Givethe exact value in simplified form. Rationalize any...

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schooledmom | (Level 1) Salutatorian

Posted June 6, 2013 at 4:39 PM via web

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Simplify the expression.  Givethe exact value in simplified form. Rationalize any demoninators.

The square root of 54

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted June 6, 2013 at 5:43 PM (Answer #1)

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You should express `54` as the following product of prime factors, such that:

`54 = 1*2*3^3`

You may evaluate the square root such that:

`sqrt 54 = sqrt(1*2*3^3) `

You should convert the square root into a rational power, such that:

`sqrt(1*2*3^3) = (1*2*3*3^2)^(1/2)`

Using the exponents properties yields:

`(1*2*3*3^2)^(1/2) = 1^(1/2)*2^(1/2)*3^(1/2)*3^(2*1/2)`

`(1*2*3*3^2)^(1/2) = 3*1*(2*3)^(1/2)`

Converting back the rational power into radical yields:

`sqrt 54 = 3sqrt 6`

Hence, evaluating `sqrt 54` using the exponents properties, yields `sqrt 54 = 3sqrt 6.`

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