Radicals and Rational Indices

If x2 = a, then x is a square root of a.

Radicals

Do you remember what square root means?

For example: 32 = 9 3 is a square root of 9.

72 = 49 7 is a square root of 49.

(3)2 = 9 3 is a square root of 9.

Note that both 3 and 3 are square roots of 9.

How about x3 = a?

If x3 = a x is a cube root of a.

If x4 = a x is a fourth root of a.

If x5 = a x is a fifth root of a.

53 = 125 5 is a cube root of 125.24 = (2)4 = 16 2 and 2 are fourth roots of 16.

105 = 100 000 10 is a fifth root of 100 0000.

For example:

If n is a positive integer such that xn = a, then x is an n th root of a.

denote n th root of a

For example:

,51253 ,2164 ,10000 0015 283

It should be noticed that: positive fourth root of 16 4 16 negative fourth root of 16

4 16

n aRadical

Find the value of .3 216

Follow-up question

216666

62163

21663

We learnt that (am)n = amn for integral indices.

Suppose the law also hold for rational indices, then

Rational Indices

How about the index is a rational number,

for example ?10

1

2

1

, aa

22

1

)(a2

2

1

a a By definition,

is the square root of a.2

1

a

33

1

)(a3

3

1

a a

aa 2

1

33

1

aa

, where n is a positive integer.n aan

1

In general,

By definition,

is the cube root of a.3

1

a

where m and n are integers and a > 0, n >0.,)( n mmn aaa

n

m

To summarize, we have

Then, what is the

meaning of ?n

m

a

n

m

a mna )(1

mn a )( nma1

)( n man

m

aand

For example,

,)10()10(10 3434

1

4

3

4 34

134

3

10)10(10

Find the value of .2

3

144

2

3

144 2

32 )12(

1728

2

3 2

)12(

312

Find the value of .4

3

81

Follow-up question

4

3

81

4

34 )3(

27

1

4

3 4

)3(33

Simplify and express

your answer with positive index.

3

3 2

x

x

2

3

3

2

x

x

6

5

x

2

3

3

2

x

6

94

x

3

3 2

x

x

6

5

1

x

Simplify and express your answer with positive index.

4 53 xx

Follow-up question

4

5

2

3

xx

4

11

x

4

5

2

3

x

4 53 xx

4

56

x