CHAPTER 8 / ESSENTIAL ALGEBRA I SKILLS 313
What Are Roots?
The Latin word radixmeans root(remember that
radishesgrow underground), so the word radical
means the root of a number(or a person who seeks to
change a system “from the roots up”). What does the
root of a plant have to do with the root of a number?
Think of a square with an area of 9 square units
sitting on the ground:
The bottom of the square is “rooted” to the
ground, and it has a length of 3. So we say that 3 is the
square rootof 9!
The square root of a number is what you must
square to get the number.
All positive numbers have two square roots. For
instance, the square roots of 9 are 3 and −3.
The radical symbol, , however, means only the
non-negativesquare root. So although the square
root of 9 equals either 3 or −3, equals only 3.
The number inside a radical is called a radicand.
Example:
If x^2 is equal to 9 or 16, then what is the least possible
value of x^3?
xis the square root of 9 or 16, so it could be −3, 3, −4,
or 4. Therefore, x^3 could be −27, 27, −64, or 64. The
least of these, of course, is −64.
Remember that does not always equal x.
It does, however, always equal |x|.
Example:
Simplify.
Don’t worry about squaring first, just remember the
rule above. It simplifies to
.
Working with Roots
Memorize the list of perfect squares:4, 9, 16,
25, 36, 49, 64, 81, 100. This will make working
with roots easier.
31 x
y
+
31
2
x
y
⎛ +
⎝⎜
⎞
⎠⎟
x^2
9
To simplify a square root expression, factor any
perfect squares from the radicand and simplify.
Example:
Simplify
Simplify
When adding or subtracting roots, treat them
like exponentials: combine only like terms—
those with the same radicand.
Example:
Simplify
When multiplying or dividing roots, multiply or
divide the coefficients and radicands separately.
Example:
Simplify.
Simplify.
You can also use the commutative and asso-
ciative laws when simplifying expressions with
radicals.
Example:
Simplify.
25 25 25 25 2 2 2 5 5 5
85 5 405
3
()=××=××()()× ×
=×× =
25
3
()
5 3xx×=×2 5^223 ()5 2 3xx×= 5 10 15x
53 25xx×^2
86
22
8
2
6
2
== 43
86
22
37 52 137 37 137 52 167 52++ = +()+= +
37 52 137++.
mm^2 m m
2
++=+ (^1025) () 5 =+ 5
mm^2 ++ 10 25.
327 39 3 39 3 3 3 3 93=×= ×=××=
327.
Lesson 4: Working with Roots