Monday, April 28, 2014 Algebra 2 GT

Objective: We will explore and define ways to simplify radical expressions.

Warm Up: Rewrite each as an exponential expression with the smallest possible base.

€

1. 729 2. 64

3. − 3125 4. − 343

€

1. 729 = 36 2. 64 = 26

3. − 3125 4. − 343

= −5( )5

= −3( )5

Monday, April 28, 2014 Algebra 2 GT

Complete the “7.1 Review” worksheet, #1 – 12 all.

Check answers to the Exp/Log Review Packet

Unit Test on Wed 4/30

Roots of Real Numbers and

Radical Expressions

Sections 7.1 and 7.4 – Facts and Examples1.

€

anindex

radical sign

radicand

2. The positive root of a number is known as the PRINCIPAL root. So, the principal fourth root of 16 is 2 (because 24 = 16).

3. What do you notice?

4. Even roots will only yield positive answers, and we ensure this by using the absolute value bars.

5. Odd roots can be positive or negative. Why is this?

Even though because we will only work with the principal roots.

€

−2( )3=−8 ∴ −83 =−2

€

−2( )2=4, 42 =2

Because

6. What operation is associated with taking a root? In other words, what is the underlying math involved in this simplification?

€

x 6y213 = x 2y7

Therefore, another way to express roots is using RATIONAL (fraction) EXPONENTS.

€

an =a1

n and

€

amn

= an

( )m

=am

n

7. Do you recall these old exponent rules?

€

am ⋅a n =

€

am+n

€

am( )n

=

€

am n

€

am

a n=

€

am−n

€

ab( )m

=

€

ambm

€

a

b

⎛ ⎝ ⎜

⎞ ⎠ ⎟m

=

€

am

bm

€

a −m =

€

1

am

8. Here’s a hint for simplifying with rational exponents: Always rewrite numbers in exponential form, using the smallest base possible.

64 = 82 = 43 =26

27 = 33

625 = 252 = 54

Definition of nth Root

** For a square root the value of n is 2.

For any real numbers a and b and any positive integers n,

if an = b, then a is the nth root of b.

Notation

indexRadical sign

radicand

Note: An index of 2 is understood but not written in a square root sign.

€

814

SimplifyTo simplify means to find x

in the equation:x4 = 81

Solution: = 3

€

814

€

814

Principal Root -The nonnegative root of a number; only use the positive value when the

square root symbol is given; however, use both square roots when you choose

to take the square root (as part of solving) or when the plus/minus symbol

is given (as shown on the next slide).

Principal square root

Opposite of principal square root

Both square roots

€

64

− 64

± 64

Examples

4

4

1. 169

2. - 8 -3

x

x

2213 x 213x

228 3 x

28 3x

Examples

323 5 x3 6

3 3 3

3. 125

4.

x

m n

25 x

33 mn mn

Taking nth roots of variable expressions:

Using absolute value signs

If the index (n) of the radical is even, and the power under the radical sign is even, yet the resulting power is odd, then we must use an absolute value sign.

an

Examples

44

626

1.

2.

an

xy

EvenEven

EvenEven

Odd

2 x yOdd

6

1826

3.

4. 3

x

yEven

Even

3 xOdd

Odd

Even

Even

2

323- y

323- y