Exponential &

Radicals KUBHEKA SN

Exponential notation

represent as to the th power .

Exponent (integers)

Base (real

number)

General case (n is any positive integers)

Special cases

Zero and negative exponent(where a c ≠ 0)

Example

Law of Exponents

Law Example

Theorem on negative Exponents

Prove:

Prove:

Example :simplifying negative exponents

(1)

8

6

682

23242

234

9

3

)()()3

1(

)3

1(

x

y

yx

yx

yx

Principal nth root Where n=positive integer greater than 1

= real number

Value for Value for

= positive real number b

Such that

=negative real number b

Such that

Properties of:

RADICALradican

d

index

Radical sign

PROPERTY EXAMPLE

Example:combining radicals

Question:

12 5

125

125

32

41

41

3 2

4

1

1

32

α

α

αα

α

α

α

α

)(

Law of Radicals

law example

WARNING!

Simplifying RadicalsOperations with

Radicals

Review - Perfect Squares

2

2

2

2

2

2

1 1

2 4

3 9

4 16

5 25

6 36

1 1

4 2

9 3

16 4

25 5

36 6

2

2

2

2

2

2

7 49

8 64

9 81

10 100

11 121

12 144

49 7

64 8

81 9

100 10

121 11

144 12

Rules for Radicals

21) a a

b) a2 ba

3) a

b b

a

Simplifying Square Roots

Simplify:

Step 1Look for Perfect Squares (Try to use the largest perfect square possible.)

Step 2Simplify Perfect Squares

Step 3Multiply the numbers inside and outside the radical separately.

48

3 16

43

4 3

If you miss the largest

perfect square, it will

just take more steps.

Simplify: 48

4 12

34

2 2 3

4 3

Variables2a a

2x xAny even power is a perfect square.

4 2

10 5

90 45

x x

x x

x x

The square root exponent is half

of the original exponent.

Odd powers

When you take the square root of an odd power, the result is always an even power and one variable left inside the radical.

5 2

11 5

91 45

x x x

x x x

x x x

Simplifying using variables

When you simplify an even power of a variable and the result is an odd power, use absolute value bars to make sure your answer is positive.

14 7

14 12 7 6

x x

x y x yEven

powers do not need absolute value.

Simplifying Numbers & VariablesSimplify: 316x

Step 1Pull out perfect squares

Step 2Simplify

16 2x x

x4 x

4x x

Radical Multiplication

a ab b You can only multiply radicals by other radicals

8 3Both under the radical

CAN multiply

8 3Not under the radical

CANNOT multiply

What is an “nth Root?”

Extends the concept of square roots.

For example:

A cube root of 8 is 2, since 23 = 8

A fourth root of 81 is 3, since 34 = 81

For integers n greater than 1, if bn = a then b is an nth root of a.

Written where n is the index of the radical.

Rational Exponents

nth roots can be written using rational exponents.

For example:

In general, for any integer n greater than 1.

Real nth Roots If n is odd:

a has one real nth root

If n is even:

And a > 0, a has two real nth roots

And a = 0, a has one nth root, 0

And a < 0, a has no real nth roots

Finding nth Roots

Find the indicated real nth root(s) of a.

Example: n = 3, a = -125

n is odd, so there is one real cube root: (-5)3 = -125

We can write

Your Turn!

Solve each equation.

5x4 = 80

(x – 1)3 = 32

http://www.slideshare.net/nurulatiyah/radical-and-exponents-2?qid=b15cb847-ee58-4b34-aaba-ce8e8ab498a5&v=default&b=&from_search=10

http://www.slideshare.net/holmsted/roots-and-radical-expressions?qid=b15cb847-ee58-4b34-aaba-ce8e8ab498a5&v=default&b=&from_search=12

http://www.slideshare.net/hisema01/71-nth-roots-and-rational-exponents?qid=b15cb847-ee58-4b34-aaba-ce8e8ab498a5&v=default&b=&from_search=15