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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 3. Solve for x:
|
(x + 3)² |
= |
5 |
|
Solution. This has the form |
|
|
z² |
= |
a. |
|
Therefore we solve it in the same manner as Example 2. |
|
|
x + 3 |
= |
± |
|
|
x |
= |
−3 ±  |
Problem 6. 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.
Problem 7. Evaluate the following.
a) |
· = 3 |
|
b) |
· = 5 |
|
c) |
( )² = a + b |
|
d) |
=  |
|
e) |
(5 )² = 25· 2 = 50 |
|
f) |
(a4 )² = a8· 3b = 3a8b |
Example 4. Multiply out  ( + ). That is, distribute  .
Solution. |
 ( + ) |
= |
 · +  ·  |
|
|
= |
2 + 3 |
Problem 8. 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 5. 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, any fraction |
a b |
can be written as the numerator times the |
reciprocal of the denominator.
Problem 10. Rationalize the denominator: |
2
 |

Problem 11. Show each of the following by transforming the left-hand side.
Real numbers
A real number is any number that you would expect to find on the number line. The absolute value of each real number names a distance from 0.

The two main categories of real numbers are rational and irrational. These 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, 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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