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An n th root ( ) of b is a solution of the equation . Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If b<0, no real roots. If b>0, or b<0, there is exactly one real root. b x n n b 4 16 4 16 3 27 3 27

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Page 1: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

An nth root ( ) of b is a solution of the equation .

Roots of Real Numbers

bxn n b

n is even n is odd

If b>0, two real roots.Principal root is positive.

If b<0, no real roots.

4 16

4 16

If b>0, or b<0, there is exactly one real root.

3 27 3 27

Page 2: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Simplify the following roots:

Roots of Real Numbers

5 2433 64

3 8 4 16

Page 3: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

What is the difference between the following expressions?

Roots of Real Numbers

4 42 3 32

Page 4: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

What is the difference between the following expressions?

Roots of Real Numbers

44 2 33 2

Page 5: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Simplify the following:

Roots of Real Numbers

5 53 44 2

33 4 4 45

Page 6: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Learning Log Summary

LT 1 – I can identify the roots of real numbers.

The nth root is…

If a negative number is in an odd/even root…

Page 7: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Closure

Homework (record on Learning Log):

LT 1: pg. 262 ~ 1-13 (Odd)pg. 263 ~ Mixed Review

1-5

Page 8: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Consider:

and

Complete the following property of radicals:

Simplifying Radicals

63694 6329494

n ab

Page 9: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Consider:

and

Complete the following property of radicals:

Simplifying Radicals

3

2

9

4

3

2

9

4

9

4

n

b

a

Page 10: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

The properties we just discovered DO NOT apply to addition and subtraction.

CAUTION!!!!

9494

Page 11: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Simplify:

Simplifying Radicals

98 3

8

81

33 1025 5

60

Page 12: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Rationalizing the Denominator means to re-write a fraction with a radical in the denominator so that the only radical is in the numerator.

To accomplish this, multiply by some form of 1 so that the denominator can be simplified without a radical.

Ex 1)

Ex 2)

Simplifying Radicals

3

5

3

4

c

Page 13: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Learning Log Summary

LT 2 – I can simplify expressions involving radicals.

To simplify a number without the given root…

Rationalizing the denominator means…

Page 14: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Closure

Homework (record on Learning Log):

LT 2: pg. 267 ~ 1-31 (O), 39-45 (O)

Page 15: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

CONNECT 4

• Place any of the items from the answer bank into any space of your Connect 4 board without repeating.

• You will not use all of the answers!

• Solve each problem, and if the answer is on the board, place an X in the space.

• First three people to Connect 4 win!

Page 16: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Connect 4

Simplify: 62

Page 17: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Connect 4

Simplify: 98

Page 18: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Connect 4

Simplify: 3 40

Page 19: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Connect 4

Simplify: 2

2

Page 20: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Connect 4

Simplify: 3 4

2

Page 21: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Connect 4

Simplify:

3

3

2

48

Page 22: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Connect 4

Simplify: 600

Page 23: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Connect 4

Simplify: 3

27

16

Page 24: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Connect 4

Simplify: 4 32

Page 25: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Connect 4

Simplify: 75

Page 26: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Connect 4

Simplify:

3

2

Page 27: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Connect 4

Simplify:

7

56

Page 28: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Connect 4

Simplify: 33 88

Page 29: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Connect 4

Simplify: 1510

Page 30: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Connect 4

Simplify:

16

3

Page 31: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Checkpoint #1

Page 32: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Checkpoint #2

Page 33: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Checkpoint #3

Page 34: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Checkpoint #4

Page 35: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Learning Log Summary

LT 3 – I can simplify expressions involving the sum of radicals.

In order to add or subtract radicals…

Roots that can be added or subtracted…

Page 36: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Closure

Homework (record on Learning Log):

LT 3: pg. 272 ~ 1-25 (O)

Page 37: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Multiplying / Dividing Radical Expressions

Simplify the following:

72374

Page 38: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Multiplying / Dividing Radical Expressions

Simplify the following:

2632

Page 39: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Multiplying / Dividing Radical Expressions

Simplify the following:

23542354

Page 40: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Conjugates

Notice that the value of is an integer.

Expressions of the form and are called conjugates. Multiplying by a conjugate “eliminates” the radicals.

23542354

dcba dcba

Page 41: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Multiplying / Dividing Radical Expressions

Simplify the following:

53

53

Page 42: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Multiplying / Dividing Radical Expressions

Simplify the following:

37

372

Page 43: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Learning Log Summary

LT 4 – I can simplify the product and quotients of binomials that contain radicals.

When multiplying radicals…

If the denominator of a fraction contains a radical expression…

Page 44: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Closure

Homework (record on Learning Log):

LT 4: pg. 275 ~ 1-29 (O)

Page 45: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Solving Radical Equations

Any equation that contains a variable in the radicand is called a radical equation. For example:

As a non-example:

312 x

523 xx

Page 46: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Solving Radical Equations

REVIEW – What is an extraneous solution?

For Example: 3x

Page 47: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Solving Radical Equations

Solve: 253 xx

Page 48: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Solving Radical Equations

Solve: 12252 xx

Page 49: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Learning Log Summary

LT 5 – I can solve equations involving radicals.

To solve a radical equation…

LT 6 – I can identify extraneous solutions to radical equations, using appropriate justifications.

An extraneous solution is…

Page 50: An n th root ( ) of b is a solution of the equation. Roots of Real Numbers n is even n is odd If b>0, two real roots. Principal root is positive. If

Closure

Homework (record on Learning Log):

LT 5/6: pg. 280 ~ 1-19 (O) pg. 282 ~ 1-12 (all)

QUIZ on MONDAY on LT 1-6


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