Lesson Amazing calculations with fractions that contain quadratic irrationalities in denominators

Algebra ->  Polynomials-and-rational-expressions -> Lesson Amazing calculations with fractions that contain quadratic irrationalities in denominators      Log On


   

This Lesson (Amazing calculations with fractions that contain quadratic irrationalities in denominators) was created by by ikleyn(52754) About Me : View Source, Show
About ikleyn:

Amazing calculations with fractions that contain quadratic irrationalities in denominators


In this lesson you will learn,  via amazing examples,  how elimination of quadratic irrationalities in denominators of fractions allows to simplify and easily calculate the values of some complicated expressions.  Problems  1,  2,  3  and  4  below are the first set of examples.

Problem 1

Simplify the fraction  1%2F%28sqrt%282%29+%2B+1%29.

Solution

It is simple.  Simply multiply the numerator and denominator by the factor  sqrt%282%29-1  to exclude irrationality in the denominator of the original fraction:

1%2F%28sqrt%282%29+%2B+1%29 = %28sqrt%282%29-1%29%2F%28%28sqrt%282%29%2B1%29%2A%28sqrt%282%29-1%29%29 = %28sqrt%282%29-1%29%2F%28sqrt%282%29%5E2-1%29 = %28sqrt%282%29-1%29%2F%282-1%29 = sqrt%282%29-1.

Answer.  1%2F%28sqrt%282%29+%2B+1%29 = sqrt%282%29-1.


Problem 2

Simplify and find the value of the sum of fractions  1%2F%28sqrt%282%29+%2B+1%29 + 1%2F%28sqrt%283%29+%2B+sqrt%282%29%29.

Solution

Again,  it is not too hard.  Simply multiply the numerator and denominator of each fraction by a factor to exclude irrationalities in the denominators.
The factor is  sqrt%282%29-1  for the first fraction,  and  sqrt%283%29-sqrt%282%29  for the second fraction.  Then cancel the like terms that have the opposite signs.

1%2F%28sqrt%282%29+%2B+1%29 + 1%2F%28sqrt%283%29+%2B+sqrt%282%29%29 = %28sqrt%282%29-1%29%2F%28%28sqrt%282%29%2B1%29%2A%28sqrt%282%29-1%29%29 + = %28sqrt%282%29-1%29%2F%28sqrt%282%29%5E2-1%5E2%29 + %28sqrt%283%29-sqrt%282%29%29%2F%28sqrt%283%29%5E2-sqrt%282%29%5E2%29 = %28sqrt%282%29-1%29%2F%282-1%29 + %28sqrt%283%29-sqrt%282%29%29%2F%283-2%29 = %28sqrt%282%29-1%29%2F1 + %28sqrt%283%29-sqrt%282%29%29%2F1 = sqrt%282%29-1 + sqrt%283%29-sqrt%282%29 = sqrt%283%29+-+1.

Answer.  1%2F%28sqrt%282%29+%2B+1%29 + 1%2F%28sqrt%283%29+%2B+sqrt%282%29%29 = sqrt%283%29+-+1 = 0.732 (approximately).


Problem 3

Simplify and find the value of the sum of fractions  1%2F%28sqrt%282%29+%2B+1%29 + 1%2F%28sqrt%283%29+%2B+sqrt%282%29%29 + 1%2F%28sqrt%284%29+%2B+sqrt%283%29%29.

Solution

Apply the same method as in the previous  Problem 2:  multiply the numerator and denominator of each fraction by a factor to exclude irrationalities in the denominators.
The factor is  sqrt%282%29-1  for the first fraction,  sqrt%283%29-sqrt%282%29  for the second fraction,  and  sqrt%284%29-sqrt%283%29  for the third fraction.  Then cancel the like terms that have the opposite signs.

1%2F%28sqrt%282%29+%2B+1%29 + 1%2F%28sqrt%283%29+%2B+sqrt%282%29%29 + 1%2F%28sqrt%284%29+%2B+sqrt%283%29%29 = %28sqrt%282%29-1%29%2F%28%28sqrt%282%29%2B1%29%2A%28sqrt%282%29-1%29%29 + + = %28sqrt%282%29-1%29%2F%28sqrt%282%29%5E2-1%5E2%29 + %28sqrt%283%29-sqrt%282%29%29%2F%28sqrt%283%29%5E2-sqrt%282%29%5E2%29 + %28sqrt%284%29-sqrt%283%29%29%2F%28sqrt%284%29%5E2-sqrt%283%29%5E2%29 = %28sqrt%282%29-1%29%2F%282-1%29 + %28sqrt%283%29-sqrt%282%29%29%2F%283-2%29 + %28sqrt%284%29-sqrt%283%29%29%2F%284-3%29 = sqrt%282%29-1 + sqrt%283%29-sqrt%282%29 + sqrt%284%29-sqrt%283%29 = sqrt%284%29+-+1 = 2-1 = 1.

Answer.  1%2F%28sqrt%282%29+%2B+1%29 + 1%2F%28sqrt%283%29+%2B+sqrt%282%29%29 + 1%2F%28sqrt%284%29+%2B+sqrt%283%29%29 = sqrt%284%29+-+1 = 2 - 1 = 1.


Problem 4

Simplify the sum of fractions  1%2F%28sqrt%282%29+%2B+1%29 + 1%2F%28sqrt%283%29+%2B+sqrt%282%29%29 + 1%2F%28sqrt%284%29+%2B+sqrt%283%29%29 + . . . + 1%2F%28sqrt%28n%29+%2B+sqrt%28n-1%29%29.

Solution.

Apply the same method as in the previous  Problem 3:  multiply the numerator and denominator of each fraction by a factor to exclude irrationalities in the denominators.
The factor is  sqrt%282%29-1  for the first fraction,  sqrt%283%29-sqrt%282%29  for the second fraction,  sqrt%284%29-sqrt%283%29  for the third fraction,  and so on.  Then cancel the like terms that have the opposite signs.

1%2F%28sqrt%282%29+%2B+1%29 + 1%2F%28sqrt%283%29+%2B+sqrt%282%29%29 + 1%2F%28sqrt%284%29+%2B+sqrt%283%29%29 + . . . + 1%2F%28sqrt%28n%29+%2B+sqrt%28n-1%29%29 = %28sqrt%282%29-1%29%2F%28%28sqrt%282%29%2B1%29%2A%28sqrt%282%29-1%29%29 + + + . . . + =

= %28sqrt%282%29-1%29%2F%28sqrt%282%29%5E2-1%5E2%29 + %28sqrt%283%29-sqrt%282%29%29%2F%28sqrt%283%29%5E2-sqrt%282%29%5E2%29 + %28sqrt%284%29-sqrt%283%29%29%2F%28sqrt%284%29%5E2-sqrt%283%29%5E2%29 + . . . + %28sqrt%28n%29-sqrt%28n-1%29%29%2F%28sqrt%28n%29%5E2+-+sqrt%28n-1%29%5E2%29 = %28sqrt%282%29-1%29%2F%282-1%29 + %28sqrt%283%29-sqrt%282%29%29%2F%283-2%29 + %28sqrt%284%29-sqrt%283%29%29%2F%284-3%29 + . . . + %28sqrt%28n%29-sqrt%28n-1%29%29%2F%28n-%28n-1%29%29 = sqrt%282%29-1 + sqrt%283%29-sqrt%282%29 + sqrt%284%29-sqrt%283%29 + . . . + sqrt%28n%29-sqrt%28n-1%29= sqrt%28n%29+-+1.

Answer.  1%2F%28sqrt%282%29+%2B+1%29 + 1%2F%28sqrt%283%29+%2B+sqrt%282%29%29 + 1%2F%28sqrt%284%29+%2B+sqrt%283%29%29 + . . . + 1%2F%28sqrt%28n%29+%2B+sqrt%28n-1%29%29 = sqrt%28n%29+-+1.



Problems  5,  6,  7  and  8  below are the second set of examples.


Problem 5

Simplify the fraction  1%2F%28sqrt%283%29+%2B+1%29.

Solution

It is simple.  Simply multiply the numerator and denominator by the factor  sqrt%283%29-1  to exclude irrationality in the denominator of the original fraction:

1%2F%28sqrt%283%29+%2B+1%29 = %28sqrt%283%29-1%29%2F%28%28sqrt%283%29%2B1%29%2A%28sqrt%283%29-1%29%29 = %28sqrt%283%29-1%29%2F%28sqrt%283%29%5E2-1%29 = %28sqrt%283%29-1%29%2F%283-1%29 = %28sqrt%283%29-1%29%2F2.

Answer.  1%2F%28sqrt%283%29+%2B+1%29 = %28sqrt%283%29-1%29%2F2.


Problem 6

Simplify and find the value of the sum of fractions  1%2F%28sqrt%283%29+%2B+1%29 + 1%2F%28sqrt%285%29+%2B+sqrt%283%29%29.

Solution

Again,  it is not too hard.  Simply multiply the numerator and denominator of each fraction by a factor to exclude irrationalities in the denominators.
The factor is  sqrt%283%29-1  for the first fraction,  and  sqrt%285%29-sqrt%283%29  for the second fraction.  Then cancel the like terms that have the opposite signs.

1%2F%28sqrt%283%29+%2B+1%29 + 1%2F%28sqrt%285%29+%2B+sqrt%283%29%29 = %28sqrt%283%29-1%29%2F%28%28sqrt%283%29%2B1%29%2A%28sqrt%283%29-1%29%29 + = %28sqrt%283%29-1%29%2F%28sqrt%283%29%5E2-1%5E2%29 + %28sqrt%285%29-sqrt%283%29%29%2F%28sqrt%285%29%5E2-sqrt%283%29%5E2%29 = %28sqrt%283%29-1%29%2F%283-1%29 + %28sqrt%285%29-sqrt%283%29%29%2F%285-3%29 = %28sqrt%283%29-1%29%2F2 + %28sqrt%285%29-sqrt%283%29%29%2F2 = 1%2F2.(sqrt%283%29-1 + sqrt%285%29-sqrt%283%29) = 1%2F2.(sqrt%285%29+-+1).

Answer.  1%2F%28sqrt%283%29+%2B+1%29 + 1%2F%28sqrt%285%29+%2B+sqrt%283%29%29 = 1%2F2.(sqrt%285%29+-+1) = 0.618 (approximately).


Problem 7

Simplify and find the value of the sum of fractions  1%2F%28sqrt%283%29+%2B+1%29 + 1%2F%28sqrt%285%29+%2B+sqrt%283%29%29 + 1%2F%28sqrt%287%29+%2B+sqrt%285%29%29.

Solution

Apply the same method as in the previous  Problem 6:  multiply the numerator and denominator of each fraction by a factor to exclude irrationalities in the denominators.
The factor is  sqrt%283%29-1  for the first fraction,  sqrt%285%29-sqrt%283%29  for the second fraction,  and  sqrt%287%29-sqrt%285%29  for the third fraction.  Then cancel the like terms that have the opposite signs.

1%2F%28sqrt%283%29+%2B+1%29 + 1%2F%28sqrt%285%29+%2B+sqrt%283%29%29 + 1%2F%28sqrt%287%29+%2B+sqrt%285%29%29 = %28sqrt%283%29-1%29%2F%28%28sqrt%283%29%2B1%29%2A%28sqrt%283%29-1%29%29 + + = %28sqrt%283%29-1%29%2F%28sqrt%283%29%5E2-1%5E2%29 + %28sqrt%285%29-sqrt%283%29%29%2F%28sqrt%285%29%5E2-sqrt%283%29%5E2%29 + %28sqrt%287%29-sqrt%285%29%29%2F%28sqrt%287%29%5E2-sqrt%285%29%5E2%29 = %28sqrt%283%29-1%29%2F%283-1%29 + %28sqrt%285%29-sqrt%283%29%29%2F%285-3%29 + %28sqrt%287%29-sqrt%285%29%29%2F%287-5%29 = 1%2F2.(sqrt%283%29-1 + sqrt%285%29-sqrt%283%29 + sqrt%287%29-sqrt%285%29) = 1%2F2.(sqrt%287%29+-+1).

Answer.  1%2F%28sqrt%283%29+%2B+1%29 + 1%2F%28sqrt%285%29+%2B+sqrt%283%29%29 + 1%2F%28sqrt%287%29+%2B+sqrt%285%29%29 = 1%2F2.(sqrt%287%29+-+1) = 0.823 (approximately).


Problem 8

Simplify the sum of fractions  1%2F%28sqrt%283%29+%2B+1%29 + 1%2F%28sqrt%285%29+%2B+sqrt%283%29%29 + 1%2F%28sqrt%287%29+%2B+sqrt%285%29%29 + . . . + 1%2F%28sqrt%282n%2B1%29+%2B+sqrt%282n-1%29%29.

Solution.

Apply the same method as in the previous  Problem 7:  multiply the numerator and denominator of each fraction by a factor to exclude irrationalities in the denominators.
The factor is  sqrt%283%29-1  for the first fraction,  sqrt%285%29-sqrt%283%29  for the second fraction,  sqrt%287%29-sqrt%285%29  for the third fraction,  and so on.  Then cancel the like terms that have the opposite signs.

1%2F%28sqrt%283%29+%2B+1%29 + 1%2F%28sqrt%285%29+%2B+sqrt%283%29%29 + 1%2F%28sqrt%287%29+%2B+sqrt%285%29%29 + . . . + 1%2F%28sqrt%282n%2B1%29+%2B+sqrt%282n-1%29%29 = %28sqrt%283%29-1%29%2F%28%28sqrt%283%29%2B1%29%2A%28sqrt%283%29-1%29%29 + + + . . . + =

= %28sqrt%283%29-1%29%2F%28sqrt%283%29%5E2-1%5E2%29 + %28sqrt%285%29-sqrt%283%29%29%2F%28sqrt%285%29%5E2-sqrt%283%29%5E2%29 + %28sqrt%287%29-sqrt%285%29%29%2F%28sqrt%287%29%5E2-sqrt%285%29%5E2%29 + . . . + %28sqrt%282n%2B1%29-sqrt%282n-1%29%29%2F%28sqrt%282n%2B1%29%5E2+-+sqrt%282n-1%29%5E2%29 = %28sqrt%283%29-1%29%2F%283-1%29 + %28sqrt%285%29-sqrt%283%29%29%2F%283-11%29 + %28sqrt%287%29-sqrt%285%29%29%2F%287-5%29 + . . . + %28sqrt%282n%2B1%29-sqrt%282n-1%29%29%2F%282n%2B1-%282n-1%29%29 = 1%2F2.(sqrt%283%29-1 + sqrt%285%29-sqrt%283%29 + sqrt%287%29-sqrt%285%29 + . . . + sqrt%282n%2B1%29-sqrt%282n-1%29) = 1%2F2.(sqrt%282n%2B1%29+-+1).

Answer.  1%2F%28sqrt%283%29+%2B+1%29 + 1%2F%28sqrt%285%29+%2B+sqrt%283%29%29 + 1%2F%28sqrt%287%29+%2B+sqrt%285%29%29 + . . . + 1%2F%28sqrt%282n%2B1%29+%2B+sqrt%282n-1%29%29 = 1%2F2.(sqrt%282n%2B1%29+-+1).



Solve yourself the next problem.


Problem 9

Simplify the sum of fractions  1%2F%28sqrt%284%29+%2B+1%29 + 1%2F%28sqrt%287%29+%2B+sqrt%284%29%29 + 1%2F%28sqrt%2810%29+%2B+sqrt%287%29%29 + . . . + 1%2F%28sqrt%283n%2B1%29+%2B+sqrt%283%28n-1%29%2B1%29%29.


My other closely related lessons in this site are
    - HOW TO rationalize a fraction by making its denominator free of square roots
    - HOW TO rationalize a fraction by making its denominator free of cubic roots
    - OVERVIEW of lessons on simplifying and rationalizing expressions and functions (denominators)

Use this file/link  ALGEBRA-I - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-I.

This lesson has been accessed 2340 times.