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Rationalize the Denominator (7√(2) + √(6)) / (14√(2) + √(6))

by Michelle Campo
(Gretna, La, US )











































   • Rationalize the Denominator

           Simplify the following expression by so that the denominator contains no square root symbols.

           (7√(2) + √(6)) / (14√(2) + √(6))

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Nov 03, 2012
Rationalize the Denominator
by: Staff

Answer

Part I

Rationalizing the Denominator is the standard way of simplifying fractions containing radicals in the denominator. Rationalizing the denominator means to “rewrite the fraction so there are no radicals in the denominator”.

Your problem: rationalize the denominator of the following fraction:

(7√(2) + √(6)) / (14√(2) + √(6))

Your problem has two terms in the denominator: a + b

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 by multiplying the denominator by its conjugate: a - b.


“IF” the original denominator [14√(2) + √(6)] could by multiplied by its conjugate [14√(2) - √(6)], then both √ signs in the denominator would disappear, since:

[14√(2) + √(6)] * [14√(2) - √(6)]

= [14√(2)]² - [√(6)]²

= 14 * 14 * √(2) * √(2) - [√(6) * √(6)]

= 14 * 14 * 2 - 6

= 392 - 6

= 386


However, in order to preserve the value of the original fraction, both the numerator and denominator must each be multiplied by the same amount: [14√(2) - √(6)].

To apply this concept, multiply the original fraction by [14√(2) - √(6)] / [14√(2) - √(6)]. The fraction [14√(2) - √(6)] / [14√(2) - √(6)] 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] * [14√(2) - √(6)] / [14√(2) - √(6)]

= [original fraction] * 1

= [original fraction]


Therefore,

= [original fraction] * [14√(2) - √(6)] / [14√(2) - √(6)]

= {[7√(2) + √(6)] / [14√(2) + √(6)]} * {[14√(2) - √(6)] / [14√(2) - √(6)]}

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Nov 03, 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:


= {[7√(2) + √(6)] * [14√(2) - √(6)]} / {[14√(2) + √(6)] * [14√(2) - √(6)]}

= [7√(2) *14*√(2) - 7√(2) *√(6) + √(6) *14*√(2) - √(6) * √(6) ] / {[14√(2)]² - [√(6)]²}

= [196 - 14√ (3) + 28√ (3) - 6 ] / {14 * 14 * √(2) * √(2) - [√(6) * √(6)]}

= [28√ (3) - 14√ (3) + 196 - 6] / (14 * 14 * 2 - 6)

= [14√ (3) + 190] / (392 - 6)

= [14√(3) + 190] / 386

= [(7√(3) + 95] / 193

= 0.5550484748859


Final Answer:

[(7√(3) + 95] / 193



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Check this answer against the original expression with a calculator:

Final answer:

[(7√(3) + 95] / 193 = 0.5550484748859


Original expression:

(7√(2) + √(6)) / (14√(2) + √(6)) = 0.5550484748859


Since the decimal value computed from the original expression and the decimal value computed from the final answer are the same, the solution is valid.



Thanks for writing.

Staff
www.solving-math-problems.com



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