Roots and Radicals
Here’s what you already know: (1) roots express fractional
exponents; (2) it’s often easier to work with roots by converting
them into expressions that look like this:
Roots and powers are reciprocals. To square the number
3,
multiply
3 by itself:
32 = 3 
3 = 9. To get the root of 9,
, you have to find the number that,
multiplied by itself, will equal
9. That number is
3.
Square roots appear far more often than any other kind
of root on the SAT, but cube roots, fourth roots, fifth roots, and
so on could conceivably make an appearance. Each root is represented
by a radical sign with the appropriate number next to it (a radical
without any superscript denotes a square root). For example, cube
roots are shown as
, fourth roots
as
, and so on. Roots of higher degree
operate the same way square roots do. Because
33 =
27, it follows that the cube root of
27 is
3.
Here are a few examples:
Adding and Subtracting Roots
You can’t add or subtract roots. You have to work out
each root separately and then add. To solve
,
do not add the
9 and 4 together to get
. Instead,
.
The SAT tests if you remember this rule by including trap
answers that do add or subtract roots.
Multiplying and Dividing Roots
If you’re multiplying or dividing two roots,
you can multiply or divide the numbers under the root sign as long
as the roots are of the same degree. You can multiply or divide
two square roots for instance, but you can’t multiply a square root and
a cube root.
