|
|
|
|
|
|
|
||
|
|
|
|
|
Other Roots and Radicals / Domains / Notes (page 2 of 4) Sections: Square roots, Other roots / Domains, Further simplifying, Rationalizing denominators So far we have dealt only with square roots, but you would manipulate other radicals in much the same way. However, we need first to cover the notation for these other radicals. Non-square-root radicals
need a number to indicate which root is meant. By default, the simple
radical symbol, " a square (second) root
is written as a cube (third) root is
written as a fourth root is written
as a fifth root is written
as: ...and so on. Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved When writing a square root, you can put a teeny "2" in, if you like, but this would be considered non-standard notation. The process of simplification in these other roots works similarly to simplification of square roots. If you have a cube root, you can take out any factor that occurs in threes:
In a fourth root, take out any factor that occurs in fours; in a fifth root, take out any factor that occurs in fives; etc. Usually, we cannot have
a negative inside a square root. (The exception
is for "imaginary" numbers. If you haven't done the number "i"
yet, then you haven't done imaginaries.) So, for instance,
The fact that I have the expression x – 2 inside a square root requires that x – 2 be zero or greater, so I must have x – 2 > 0. Solving, I get: domain: x > 2 On the other hand, you CAN have a negative inside a cube root (or any other odd root). For instance:
...because (–2)3 = –8.
For domain: all x Do not confuse "simplifying"
with "solving". If you have "x2
= 4", and you
take the square root of either side, you will get x
= ± 2, because you
could square either of –
2 and +2
to get +4. But
...and:
Warning: Do not confuse
these. Finding all the given roots of an equation (such as finding all
the solution to the equation "x2
= 4") is different
from simplifying an expression (such as " << Previous Top | 1 | 2 | 3 | 4 | Return to Index Next >>
|
|
|
|
Copyright © 1999-2009 Elizabeth Stapel | About | Terms of Use |
|
|
|
|
|
|
",
is assumed to mean "the square root". This is similar to exponents:
if you have "x2",
you know this means "x
squared"; if you have just plain old "x",
you know that the exponent is "1".
Whenever there is no exponent, the exponent is understood to be "1";
to indicate other powers, you insert a number. In the same way, when there
is no number on the radical, the radical is understood to be the square
root; to indicate other roots, you insert a number, called the "index",
inside that little "hook" on the front of the radical symbol.
For instance:
is not possible. Do not try to say something like " 
", because it's not true: 
. You must have a positive inside the square root. For instance:
,
there is NO RESTRICTION on the value of x,
because x
– 2 is welcome to
be negative inside a cube root. Then the domain is:
, because "
"
is specifically the positive
square root. That is:
