7.6 Properties of Radicals

Definitions

The symbol is called a . b is called the . n is called the .

if n is , and if n is .

We can take an root of a negative number, but we cannot take an root of a negative number.

bn

bnn =b bnn = b

radical

radicandindex

oddeven

oddeven

State the value of the root, or state that the expression is not defined as a real number.

1.

2.

3.

1253

814

−643

Theorem

For all, a, b, , and :

1.

2.

This theorem works in reverse as well. We can use it to simplify radicals or to combine two radicals. Just remember that in order to combine two radicals they MUST have the same index!

an bn

abn = an ⋅ bn

a

bn =

an

bnb≠0( )

Simplify

4.

5.

6.

320

403

125

64

Remember, we never want a in the

denominator. So, we need to rationalize it!

radical

Simplify

7.

8.

121

3

36

254

Theorem

For all b and , and m and n positive integers,

  9. Simplify.

bn

bmn = bn( )m

8124

Theorem

For k and m integers and all b and ,

10. Simplify.

bkm

bkm = bmk = bkm

368

Simplifying Roots with Variables

To simplify a root with variables we divide the exponent on the variable by the index. The quotient is the exponent on the variable

the radical, and the remainder is the exponent on the variable the radical.

inside

outside

Simplify.

11. 12. 250x83 192a11

Simplify.

13. 14. 3

12b−3753 ⋅ 723

Simplify.

15. 16. 320

503 75a12b5

Simplify.

17. 18. −32x10y155 24ab6

z23