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.

Do the problem yourself first!

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

 Example 5. · = 2. (3a)² = 3²a²()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 :

The denominator is now rational.

 2 can also take the form ½:
 2 = ½.
 For, we can write any fraction ab 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.