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rationalize the denominator by applying the Difference of Squares formula

by Susan
(Indianapolis)











































16x squared -625 y squared all over 2radicalx - 5 radical y. the 2 and the 5 are not on top of the radical(check mark) they are squarely in front.

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Jan 26, 2012
Rationalize the Denominator
by: Staff

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Part II

Multiply both numerators and multiply both denominators, just as you would when multiplying any two fractions:



            [16x² - 625y²] * [2√(x) + 5√(y)]
= ------------------------------------------------
           [2√(x) - 5√(y)] * [2√(x) + 5√(y)]


           [16x² - 625y²] * [2√(x) + 5√(y)]
= ------------------------------------------------
                     [2√(x)]² - [5√(y)]²


           [16x² - 625y²] * [2√(x) + 5√(y)]
= ------------------------------------------------
                          4x - 25y


At this point the numerator consists of two factors: [16x² - 625y²] and [2√(x) + 5√(y)]

Factor [16x² - 625y²] to be: (4x -25y) * (4x + 25y)


      (4x -25y) * (4x + 25y) * [2√(x) + 5√(y)]
= ------------------------------------------------
                          4x - 25y

The (4x - 25y) appears in both the numerator and denominator

    (4x + 25y) * [2√(x) + 5√(y)]       (4x - 25y)
= -------------------------------- * -------------
                           1                        (4x - 25y)


The factor (4x - 25y) will cancel the (4x - 25y) in the denominator.


      (4x + 25y) * [2√(x) + 5√(y)]
= ------------------------------------------- * 1
                           1


= (4x + 25y) * [2√(x) + 5√(y)]


Expand the expression:

= 8x√(x) + 2x√(y) + 5y√(x) + 125y√(y)



>>> The final answer is: 8x√(x) + 2x√(y) + 5y√(x) + 125y√(y)





Thanks for writing.
Staff

www.solving-math-problems.com


Jan 26, 2012
Rationalize the Denominator
by: Staff

Part I

Question:

by Susan
(Indianapolis)


16x squared -625 y squared all over 2radicalx - 5 radical y. the 2 and the 5 are not on top of the radical(check mark) they are squarely in front.


Answer:

      16x² - 625y²
---------------------------
      2√(x) - 5√(y)


You can rationalize the denominator by applying the Difference of Squares formula.

The difference of squares formula states that:

(a + b)(a - b) = a² - b²


You can remove the radicals from the denominator in your problem [ 2√(x) - 5√(y) ] by multiplying the denominator by its conjugate: 2√(x) + 5√(y).


Multiplying the denominator [ 2√(x) - 5√(y) ] by its conjugate [ 2√(x) + 5√(y) ] eliminates both √ signs:

[ 2√(x) - 5√(y) ] * [ 2√(x) + 5√(y) ]

= [2√(x)]² - [5√(y)]²

= 4x - 25y

However, in order to preserve the value of the original fraction, both the numerator and denominator must each be multiplied by the same amount: [ 2√(x) + 5√(y) ].

To apply this concept, multiply the original fraction by [ 2√(x) + 5√(y) ]/[ 2√(x) + 5√(y) ]. The fraction [ 2√(x) + 5√(y) ]/ [ 2√(x) + 5√(y) ] is equal to 1, so the original fraction is merely being multiplied by 1. As you can see by the following illustration, the value of the original fraction has not been changed.


= [original fraction]

= [original fraction] * [ 2√(x) + 5√(y) ]/[ 2√(x) + 5√(y) ]

= [original fraction] * 1

= [original fraction]


Therefore,

= [original fraction] * 1

           [16x² - 625y²]
= --------------------------- * 1
           [2√(x) - 5√(y)]


           [16x² - 625y²]              [2√(x) + 5√(y)]
= --------------------------- * ------------------
           [2√(x) - 5√(y)]             [2√(x) + 5√(y)]

---------------------------------------------------

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