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Difference of Squares

by Furqan
(Herndon,VA U.S.A)











































i need a step by step instructions to understand how to do this type of factoring.

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Apr 30, 2012
Difference of Squares
by: Staff


Question:

by Furqan
(Herndon,VA, U.S.A)


i need a step by step instructions to understand how to do this type of factoring.


Answer:


The difference of squares formula states that:

(a + b)(a - b) = a² - b²

(a + b) and (a - b) are called conjugates.

This idea can be applied to factoring or rationalizing the denominator.


Factoring: Expressions with exponents.

(a + b)(a - b) = a² - b²

(a² + b²)(a² - b²) = a⁴ - b⁴

(a⁴ + b⁴)(a⁴ - b⁴) = a⁸ - b⁸
.
.
.
And so on


Factoring: roots (fractional exponents):

Square roots

(a + √b)(a − √b) = a² - b

Fourth roots

[a + b^(1/4)][a - b^(1/4)] = a² - √b
.
.
.
And so on



How to rationalize the denominator of a fraction using the difference of squares:

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”.

An example problem:

7/(7 - √7)


“IF” the denominator could by multiplied by its conjugate [which is (7 + √7)], then the √ sign in the denominator would disappear, since:

(7 - √7) * (7 + √7) = 7² - (√7)² = 49 - 7 = 42


However, in order to preserve the value of the original fraction, both the numerator and denominator must each be multiplied by the same amount: (7 + √7).

To apply this concept, multiply the original fraction by (7 + √7)/(7 + √7). The fraction (7 + √7)/(7 + √7) is equal to 1, so the original fraction is merely being multiplied by 1. As you can see by the following illustration, its value has not been changed.


= [original fraction]

= [original fraction] * [(7 + √7)/(7 + √7)]

= [original fraction] * 1

= [original fraction]


Therefore,

= [original fraction] * [(7 + √7)/(7 + √7)]

= [7/(7 - √7)] * [(7 + √7)/(7 + √7)]

= 7*(7 + √7) / [(7 - √7) * (7 + √7)]

= 7 * (7 + √7) / [7² - (√7)²]

= 7 * (7 + √7) / (49 – 7)

= 7 * (7 + √7) / (42)


Reduce this fraction to its lowest terms

= 7 * (7 + √7) / (6 * 7)

= [(7 + √7) / 6 ] * (7 / 7)

= [(7 + √7) / 6 ] * (1)

= (7 + √7) / 6


The final answer is:

7/(7 - √7) = (7 + √7) / 6





Thanks for writing.

Staff
www.solving-math-problems.com



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