1. 6-1 Nth Roots
Objective: To simplify radicals
having various indices, and to use a
calculator to estimate the roots of
numbers.

2. Square Roots
What power is a square root?
A square is the inverse of a square root…
3
3 ?

3. Square Root*
Definition: For any real numbers a and b, if
2
a b then a is a square root of b or
b a
We can also write square roots using the ½
power.
1
2
b
b

4. Cube Root*
Definition: For any real numbers a and b, if
3
a b then a is a cube root of b or
3
b a
We can also write cube roots using the 1/3
power.
1
3
3
b
b

5. nth Root*
Definition: For any real numbers a and b, if
n
a b then a is a nth root of b or
n
b a
1
We can also write nth roots using the power.
n
b
1
n
n
b

6. Examples: Roots (of powers of 2)
Even Roots:
Odd Roots:
4 2
4
6
8
3
8
16 2
5
32
64 2
7
128
2
256 2
9
512
2
2
2

7. Roots of negative numbers*
Even roots: Negative numbers have no
even roots. (undefined)
Odd Roots: Negative numbers have
negative roots.
4
3
27
undefined
3

8. Examples: Roots (of powers of 2)
Even Roots:
4
undef .
Odd Roots:
3
8
2
4
16
undef .
5
32
6
64
undef .
7
128
2
9
512
2
8
256
undef .
2

9. Roots: Number and Types
Even Roots
Positive
2 (one positive, one 1 (positive)
negative)
64
Negative
Odd Roots
0 (undefined)
64
3
8
undef .
64
4
1 (negative)
3
64
4

10. MORE EXAMPLES
if n (index) is an even integer
if n is an odd integer
a<0 has no real nth roots
a<0 has one real nth root
2 16
a=0 has one real nth root
4i (not a realsolution)
3 8
a=0 has one real nth root
2
a>0 has two possible real nth roots
a>0 has one real nth roots
40
4
x
32
4 32
30
0
24 2
0
3 27
3

11. Odd Roots (of variable
expressions)*
When evaluating odd roots (n is odd) do not
use absolute values.
3
5
a
3
3
a
15
a
a
3
243 7
5
32
2

12. Evaluating Roots of Monomials
To evaluate nth roots of monomials:
(where c is the coefficient, and x, y and z are
variable expressions)
n
cxyz
n
c
n
1
n
x
n
1
n
y
n
1
n
z
(c ) ( x ) ( y ) ( z )
1
n
or
• Simplify coefficients (if possible)
• For variables, evaluate each variable separately

13. Evaluating Roots of Monomials*
To find a root of a monomial
• Split the monomial into a product of the factors,
and evaluate the root of each factor.
• Variables: divide the power by the root
Coefficients: re-write the number as a product
of prime numbers with powers, then divide the
powers by the root.
49 x 8
5
32 x10 y15
49 x 8
5
25
72
5
( x 2 )5
( x 4 )2
5
7x4
( y 3 )5
2x2 y3