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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  =  ±radicals.

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  =  ±radicals
 
  x  =  −3 ± radicals.

Problem 7.   Solve for x.

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

The definition of the square root radical

Here is the formal rule that implicitly defines the symbol radicals:

radicals

A square root radical multiplied by itself
produces the radicand.

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

Problem 8.   Evaluate the following.

   a)   radicals· radicals  =  3   b)   (radicals)²  =  5
 
  c)   (radicals)²  =  a + b   d)   radicals  =  radicals
 
  e)   (5radicals)²  =  25· 2 = 50   f)   (a4radicals)²  =  a8· 3b = 3a8b

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

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

Problem 9.   Following the previous Example, multiply out

radicalsradicals(radicals  + radicals).

radicalsradicals(radicals  + radicals) = radicals
 
  = radicals

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 
radicals

Solution.  Multiply both the numerator and denominator by radicals:

Square root of 2 over 2.

The denominator is now rational.

radicals
  2 
 can also take the form ½radicals:
radicals
  2 
  = ½radicals.
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 radicals, for example, is approximately 1.414, then we can easily know that

radicals = ½radicalsradicals½(1.414) = 0.707.
   Problem 10.   Rationalize the denominator:     2 
radicals

radicals

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

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

Real numbers

radicals

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,  radicals,  radicals,  radicals.

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 radicals.)

If radicals 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)   radicals   x − 3 0; that is, x 3.

b)   radicals    1 + x 0;  x −1.

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

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

end

Next Lesson:  Simplifying radicals

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