1. The Radical Square Root
The square root of any real number is a number, rational or irrational, that when multiplied by itself will result in a product that is the original number
The Radical
Radical sign
Square Root
Radicand
Every positive radicand has a positive and negative sq. root.
The principal Sq. Root of a number is the positive sq. root.
A rational number can have a rational or irrational sq. rt.
An irrational number can only have an irrational root.

2. =  7.9 =  232 225 = +15 529 = +23 Model Problems
Find to the nearest tenth:
= 13.4
=  7.9
=  11.4
=  232
=  64.4
Find the principal Square Root:
225
= +15
529
= +23
Simplify:
= |x|
= x
= 2x8
= x + 1

3. Index of 2 Square Root Index of 2 radical sign radicand index
of a number is one of the two
equal factors whose product is
that number
Square Root
Index of 2
has an index of 2
Every positive real number
has two square roots
The principal square root of
a positive number k is its
positive square root,
If k < 0, is an imaginary
number

4. Index of 3 Cube Root Index = 3 radical sign radicand index
of a number is one of the three
equal factors whose product is
that number
has an index of 3
principal cube roots

5. nth Root
The nth root of a number (where n is any counting number) is one of n equal factors whose product is that number.
k is the radicand
n is the index
is the principal nth root of k
25 = 32
(-2)5 = -32
54 = 625

6. Index of n nth Root Index of n radical sign radicand index
of a number is one of n
equal factors whose product is
that number
nth Root
Index of n
has an index where n is any
counting number
principal odd roots
principal even roots

7. Radical Rules!
True or False: T T T

8. In general, for non-negative numbers a, b and n
Radical Rule #1
In general, for non-negative numbers a, b and n
Example:
simplified
= x4
= x3
Hint: will the index divide evenly
into the exponent of radicand term?

9. In general, for non-negative numbers a, b, and n
Radical Rule #2
True or False: If
and T T
Transitive Property
of Equality
If a = b, and b = c,
then a = c
then
In general, for non-negative numbers a, b, and n
Example:

10. Perfect Squares – Index 212
144 11
121
100 10 9 81 8 64 7 49 6 36 5 25 4 16 3 9 4 2 1

11. Perfect Square Factors
Find as many combinations of 2 factors whose product is 75
Factors that are Perfect Squares
Find as many combinations of 2 factors whose product is 128

12. Find as many combinations of 2 factors whose product is 80
Simplifying Radicals
Simplify:
answer must be in radical form.
Find as many combinations of 2 factors whose product is 80
perfect
square
comes out
from under
the radical
To simplify a radical find, if possible, 2 factors of the radicand, one of which is the largest perfect square of the radicand.
The square root of the perfect square becomes a factor of the coefficient of the radical.

13. Perfect Cubes
13 =
23 =
33 =
43 =
53 =
63 =
73 =
(x4)3 = x12
(-2y2)3 = -8y6

14. Simplifying Radicals Simplify: answer must be in radical form.
1) Factor the radicand so that the perfect power (cube) is a factor
2) Express the radical as the product of the roots of the factors
3) Simplify the radical containing the largest perfect power (cube)

15. Simplifying Radicals Simplify:
1) Change the radicand to an equivalent fraction whose denominator is a perfect power.
2) Express the radical as the quotient of two roots
3) Simplify the radical in the denominator

16. Simplify: Model Problems
KEY: Find 2 factors - one of which is the largest perfect square possible

17. Model Problems
Simplify: