Radicals - Mixed Index

8.7

Radicals - Mixed Index

Knowing that a radical has the same properties as exponents (written as a ratio)

allows us to manipulate radicals in new ways. One thing we are allowed to do is

reduce, not just the radicand, but the index as well. This is shown in the following example.

Example 1.

p

8

x6 y 2

(x6 y 2)

6

8

x y

3

4

Rewrite as raitonal exponent

1

5

Multiply exponents

2

8

Reduce each fraction

1

4

All exponents have denominator of 4, this is our new index

Our Solution

x y

p

4

x3 y

What we have done is reduced our index by dividing the index and all the exponents by the same number (2 in the previous example). If we notice a common

factor in the index and all the exponnets on every factor we can reduce by

dividing by that common factor. This is shown in the next example

Example 2.

¡Ì

a6b9c15

¡Ì

8

a2b3c5

24

Index and all exponents are divisible by 3

Our Solution

We can use the same process when there are coefficients in the problem. We will

first write the coefficient as an exponential expression so we can divide the

exponet by the common factor as well.

Example 3.

¡Ì

9

8m6n3

¡Ì

9 3 6 3

2mn

¡Ì

3

2m2n

Write 8 as 23

Index and all exponents are divisible by 3

Our Solution

We can use a very similar idea to also multiply radicals where the index does not

match. First we will consider an example using rational exponents, then identify

the pattern we can use.

1

Example 4.

¡Ì

¡Ì

3

4

ab2 a2b

2

1

3

2

(ab ) (a b)

1

3

2

3

2

4

a b a b

4

12

8

12

6

12

1

4

1

4

3

12

a b a b

¡Ì

12 4 8 6 3

abab

¡Ì

12 10 11

a b

Rewrite as rational exponents

Multiply exponents

To have one radical need a common denominator, 12

Write under a single radical with common index, 12

Add exponents

Our Solution

To combine the radicals we need a common index (just like the common denominator). We will get a common index by multiplying each index and exponent by

an integer that will allow us to build up to that desired index. This process is

shown in the next example.

Example 5.

¡Ì

¡Ì

4

6

a2b3 a2b

¡Ì

12

a6b9a4b2

¡Ì

12 10 11

a b

Common index is 12.

Multiply first index and exponents by 3, second by 2

Add exponents

Our Solution

Often after combining radicals of mixed index we will need to simplify the

resulting radical.

Example 6.

p

p

5

x3 y 4 3 x2 y

p

15

x9 y 12x10y 5

p

15 19 17

x y

p

xy 15 x4 y 2

Common index: 15.

Multiply first index and exponents by 3, second by 5

Add exponents

Simplify by dividing exponents by index, remainder is left inside

Our Solution

Just as with reducing the index, we will rewrite coefficients as exponential expressions. This will also allow us to use exponent properties to simplify.

Example 7.

p

p

3

4x2 y 4 8xy 3

p

p

3

22x2 y 4 23xy 3

p

12

24x8 y 429x3 y 9

p

12 13 11 13

2 x y

p

2y 12 2x11y

Rewrite 4 as 22 and 8 as 23

Common index: 12.

Multiply first index and exponents by 4, second by 3

Add exponents (even on the 2)

Simplify by dividing exponents by index, remainder is left inside

Our Solution

2

If there is a binomial in the radical then we need to keep that binomial together

through the entire problem.

Example 8.

p

3x(y + z) 3 9x(y + z)2

p

p

3x(y + z) 3 32x(y + z)2

p

6

33x3(y + z)334x2(y + z)4

p

6

37x5(y + z)7

p

3(y + z) 6 3x5(y + z)

p

Rewrite 9 as 32

Common index: 6. Multiply first group by 3, second by 2

Add exponents, keep (y + z) as binomial

Simplify, dividing exponent by index, remainder inside

Our Solution

The same process is used for dividing mixed index as with multilpying mixed

index. The only difference is our final answer can¡¯t have a radical over the denominator.

Example 9.

p

6

x4 y 3z 2

p

8

x7 y 2z

r

24

x16 y 12z 8

x21y 6z 3

p

24

x?5 y 6z 5

r

y 6z 5

x5

r

x7

x7

12

r

12

y 6z 5

x5

12

p

12

!

x7 y 6z 5

x

Common index is 24. Multiply first group by 4, second by 3

Subtract exponents

Negative exponent moves to denominator

Can ¡ät have denominator in radical, need 12x ¡äs, or 7 more

Multiply numerator and denominator by

¡Ì

12

x7

Our Solution

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8.7

Practice - Radicals of Mixed Index

Reduce the following radicals.

p

1) 8 16x4 y 6

3)

5)

p

12

2)

64x4 y 6z 8

4)

p

4

9x2 y 6

q

4

25x3

16x5

p

q

6

16x2

9y 4

6)

15

x9 y 12z 6

p

10

p

64x8 y 4

7)

12

x6 y 9

8)

9)

p

8

8x3 y 6

10)

11)

p

9

8x3 y 6

12)

Combine the following radicals.

¡Ì ¡Ì

13) 3 5 6

35)

15)

¡Ì ¡Ì

x 3 7y

37)

17)

¡Ì ¡Ì

x3 x?2

39)

p

5

41)

19)

21)

23)

p

4

¡Ì

4

¡Ì

x2 y xy

xy 2

2

p

3

2

x2 y

¡Ì

5

2 3

a bc a b c

¡Ì ¡Ì

4

25) a a3

27)

¡Ì

¡Ì

5 2

b b3

29)

xy 3

31)

p

¡Ì

4

33)

p

3

p

3xy 2z 4 9x3 yz 2

9ab3

p

3

p

4

25y 2

16

81x8 y 12

p

p

27a5(b + 1) 3 81a(b + 1)4

p

¡Ì

3

a2

¡Ì

4

a

p

4

x2 y 3

¡Ì

3 xy

¡Ì

¡Ì

5

ab3c

a2b3c?1

43)

p

4

(3x ? 1)3

45)

p

3

(2x + 1)2

p

5

p

5

(3x ? 1)3

(2x + 1)2

¡Ì

¡Ì

3

74 5

¡Ì ¡Ì

16) 3 y 5 3z

14)

x2 y

p

4

9x3 yz 2

4

18)

¡Ì

4

3x

¡Ì

20)

¡Ì

ab

¡Ì

5

y+4

2a2b2

22)

¡Ì

5

24)

p

6

26)

¡Ì

3

28)

¡Ì

4

30)

32)

34)

¡Ì

5

a2b3

¡Ì

6

¡Ì

3 3

a

3

ab

36)

x2 yz 2

38)

x5

40)

2

a

¡Ì

2x3 y 3

4 3 4

abc

a2b

p

5

x2 yz 3

x2

p

¡Ì

¡Ì

4

¡Ì

3

4xy 2

2

ab c

¡Ì

3

8x (y + z)5

x2

p

3

4x2(y + z)2

¡Ì

5

x

¡Ì

5

a4b2

¡Ì

3

42)

p

5

ab2

x3 y 4z 9

p

xy ?2z

44)

p

3

46)

p

4

(5 ? 3x)3

ab

p

3

p

(2 + 5x)2

p

4

(2 + 5x)

p

3

(5 ? 3x)2

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