Radicals - Rational and irrational numbers: Level 2
The definition of the square root radical
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. | |
| The solution to any equation that looks like that is | |||
| 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.
Now, in this equation
| (x + 3)² | = | 5, | |
| the argument is x + 3. Therefore, | |||
| x + 3 | = | ± |
|
| x | = | −3 ± . |
|
Problem 7. Solve for x.
To see the answer, pass your mouse over the colored area.
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) | (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
produces the radicand.
| 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 | 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
 |
= ½ ½(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 any number that you would expect to find on the number line. Its name is the "address" of a point on the number line. Its absolute value names the distance of that point from 0.
The two main categories of real numbers are rational and irrational. Those 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
Please make a donation to keep TheMathPage online.
Even $1 will help.
Copyright © 2001-2010 Lawrence Spector
Questions or comments?
E-mail: themathpage@nyc.rr.com