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Radicals - Rational and irrational numbers:  Level 2

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Equations  (x + a)² = b

The definition of the square root radical

Rationalizing a denominator

Real numbers

Example 4.   Solve for x:

  (x + 3)²  =  5.
 
   Solution.   All equations fall into certain forms, and this one has the same
  form as Example 3:
  z²  =  a.
 
         If an equation looks like that, then the solution will look like this:
 
  z  =  ±.

In other words, if we call z the argument of the equation z² = a, then in the solution, the argument is on the left.  The argument is whatever was squared.

In this equation --

  (x + 3)²  =  5,
 
-- the argument is x + 3.  Therefore,
 
  x + 3  =  ±
 
  x  =  −3 ± .

Problem 7.   Solve for x.

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To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!

  a)   (x − 1)² = 2   b)   (x + 5)² = 6
  x − 1 = ±     x + 5 = ±
  x = 1 ±     x = −5 ±
  c)   (xp = q + r
 
  xp = ±
 
  x = p ±

The definition of the square root radical

Here is the formal rule that implicitly defines the symbol :

A square root radical multiplied by itself
produces the radicand.

   Example 5. ·    =   2.
 
  (3a  =   a²()²   (Power of a product of factors)
 
    =   9a²· 10
 
    =   90a².

Problem 8.   Evaluate the following.

   a)   ·   =  3   b)   ()²  =  5
 
  c)   ()²  =  a + b   d)     =  
 
  e)   (5)²  =  25· 2 = 50   f)   (a4)²  =  a8· 3b = 3a8b

Example 6.   Multiply out  ( + ).  That is, distribute .

  Solution. ( + ) = ·  + · 
 
  = 2 + 3

Problem 9.   Following the previous Example, multiply out

(  + ).

(  + ) =
 
  =

Rationalizing a denominator

Rationalizing a denominator is a simple technique for changing an irrational denominator into a rational one.  We simply multiply the radical by itself. But then we must multiply the numerator by the same number.

  Example 7.   Rationalize this denominator:     1 

Solution.  Multiply both the numerator and denominator by :

Square root of 2 over 2.

The denominator is now rational.


  2 
 can also take the form ½:

  2 
  = ½.
For, we can write any fraction  a
b
 as the numerator times the

reciprocal of the denominator.

Finally, rationalizing the denominator simplifies the task of evaluating the fraction.  Since we know that , for example, is approximately 1.414, then we can easily know that

= ½½(1.414) = 0.707.
   Problem 10.   Rationalize the denominator:     2 

Problem 11.   Show each of the following by transforming the left-hand side.

  a)      6 
= 2.     6 
=
  3
= 2  
  b)      9 
=
  2
    9 
=
  6
=
  2
  c)      =
  ab
=
  b

Real numbers

A real number is distinguished from an imaginary or complex number. It is what we call any rational or irrational number. It is a number we expect to find on the number line. It is a number we need for measuring.  

The real numbers are the subject of calculus and of scientific measurement.

A real variable is a variable that takes on real values.

Problem 12.   Let x be a real variable, and let 3 < x < 4.  Name five values that x might have.

For example, 3.1,  3.14,  ,  ,  .

Problem 13.   If the square root is to be a real number, then the radicand may not be negative.  (There is no such real number, for example, as .)

If is to be real, then we must have  x 0.

(If you are not viewing this page with Internet Explorer 6 or Firefox 3, then your browser may not be able to display the symbol , "is greater than or equal to"; or , "is less than or equal to.")

Therefore, what values are permitted to the real variable x ?

a)      x − 3 0; that is, x 3.

b)       1 + x 0;  x −1.

c)       1 − x 0;  −x −1, which implies x 1.

d)       x² 0.  In this case, x may be any real number.


Next Lesson:  Simplifying radicals


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