6-1 Nth RootsObjective: To simplify radicals having

various indices, and to use a calculator to estimate the roots of

numbers.

Square Roots

What power is a square root?

A square is the inverse of a square root…

?33

Square Root*

Definition: For any real numbers a and b, if

then a is a square root of b or

We can also write square roots using the ½ power.

ba 2

ab

bb 2

1

Cube Root*

Definition: For any real numbers a and b, if

then a is a cube root of b or

We can also write cube roots using the 1/3 power.

ba 3

ab 3

33

1

bb

nth Root*

Definition: For any real numbers a and b, if

then a is a nth root of b or

We can also write nth roots using the power.

ban abn

nn bb 1

n

1

Examples: Roots (of powers of 2)Even Roots: Odd Roots:

2256

264

216

24

8

6

4

2512

2128

232

28

9

7

5

3

Roots of negative numbers*

Even roots: Negative numbers have no even roots. (undefined)

Odd Roots: Negative numbers have negative roots.

3 27

4

undefined

327

43

undefined

Examples: Roots (of powers of 2)Even Roots: Odd Roots:

.256

.64

.16

.4

8

6

4

undef

undef

undef

undef

2512

2128

232

28

9

7

5

3

Roots: Number and Types

Even Roots Odd Roots

Positive 2 (one positive, one negative)

1 (positive)

Negative 0 (undefined) 1 (negative)

464.64

464864

3

3

undef

if n (index) is an even integer

a<0 has no real nth roots

a=0 has one real nth root

a>0 has two possible real nth roots

if n is an odd integer

a<0 has one real nth root

a=0 has one real nth root

a>0 has one real nth roots

solution) real a(not 42 16 i 23 8

04 0 03 0

4 224 32 324

x 33 27

MORE EXAMPLES

Odd Roots (of variable expressions)*

When evaluating odd roots (n is odd) do not use absolute values.

232

7243

535 15

33 3

aa

aa

Evaluating Roots of Monomials

To evaluate nth roots of monomials:

(where c is the coefficient, and x, y and z are variable expressions)

or

• Simplify coefficients (if possible)• For variables, evaluate each variable separately

nnnn

nnnnn

zyxc

zyxccxyz1111

)()()()(

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.

325 535 525 55 1510

424288

2)()(232

7)(74949

yxyxyx

xxxx

 

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