SAT Mc Graw Hill 2011

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