Warm-up Problems

Simplify the following:

1. (6v2 + 2v – 5) + (3v – 4v2 +7) =

2. - 4v2 (v + 1) =

3. (4n2 + 1) - (2n2 + 6n - 3) =

4. (5n+1)(n−3) =

5. (3a+2)(5a2 −a+1) =

Warm-up Problems

Solutions:

1. 2v2 + 5v + 2

2. - 4v3 – 4v2

3. 2n2 - 6n + 4

4. 15n2 - 14n – 3

5. 15a3 + 7a2 + a + 2

Introduction to Irrational Numbers

Irrational Numbers

We write “irrational numbers” using a “radical symbol,” often just referred to as a “radical.”

√3 is an irrational number.

√ is the radical sign / symbol, and 3 is called the radicand.

HOW DO WE SIMPLIFY RADICALS?

1. simplify square roots, and

2. simplify radical expressions.

In the expression , is the radical sign and

64 is the radicand.

If x2 = y then x is a square root of y.

1. Find the square root:8

2. Find the square root:-0.2

64

64

0.04

3. Find the square root: 11, -11

4. Find the square root:21

5. Find the square root:

121

441

25

815

9

6. Use a calculator to find each square root. Round the decimal

answer to the nearest hundredth.

6.82, -6.82

46.5

What numbers are perfect squares?1 • 1 = 12 • 2 = 43 • 3 = 9

4 • 4 = 165 • 5 = 256 • 6 = 36

49, 64, 81, 100, 121, 144, ...

The square root of 4 is 2

The square root of 9 is 3

The square root of 16 is 4

The square root of 25 is 5

1. Simplify Find a perfect square that goes into 147.

147

147 349

147 349

147 7 3

2. Simplify

Find a perfect square that goes into 605. 605

121 5

121 5

11 5

Simplify

A. .

B. .

C. .

D. .

2 18

72

3 8

6 236 2

Now you try some…

Now you try…

How do you simplify variables in the radical?

Look at these examples and try to find the pattern… x7

1x x2x x3x x x4 2x x5 2x x x6 3x x

What is the answer to ? x7

7 3x x x

As a general rule, divide the exponent by two. The remainder stays in the

radical.

4. Simplify Find a perfect square that goes into 49.

49x2

249 x7x

5. Simplify 258x254 2x

122 2x x

Simplify

1. 3x6

2. 3x18

3. 9x6

4. 9x18

369x

Now you try some…

Adding & Subtracting Radicals

• The rules for adding and subtracting radicals are very similar to the rules for adding and subtracting polynomials.

• The radical / radicand have to match exactly in order to add or subtract.

Ex. 1: 112116113

112116113

11)263(

117

Ex. 2:

21714 24237579

2)43(7)59(

2714

75232479

75232479

Now you try some…

IMPORTANT

• If each radical in a radical expression is not in simplest form, simplify them first.

• Then use the distributive property, whenever possible, to further simplify the expression.

Ex. 4:

752325987 325221652497

352245277

310269 310220249

752325987

Ex. 5: Simplify, then use a calculator to verify your answer.

28273 74273

72273

7473

71

65.2;7

485273 3165393

345333 32039

Ex. 6: Simplify, then use a calculator to verify your answer.

23.50;329

Now you try some…

Rational or Irrational? Justify your answer.