ROOTS and

POWERS

Rational numbers, irrational numbers

CHAPTER 4

THE REAL NUMBER SYSTEM

Natural Numbers: N = { 1, 2, 3, …}Whole Numbers: W = { 0, 1, 2 , 3, ...}Integers: I = {….. -3, -2, -1, 0, 1, 2, 3, ...}

Rational Numbers: Q a

b| a,b I ,b 0

Irrational Numbers: Q = {non-terminating, non-repeating decimals} π, e ,√2 , √ 3 ...Real Numbers: R = {all rational and irrational}

Imaginary Numbers: i = {square roots of negative numbers}

Complex Numbers: C = { real and imaginary numbers}

Natural Numbers

Whole Numbers

Integers

Rational Numbers

Irrational Num

bers

Real NumbersIm

aginary Num

bers

Complex Numbers

1.1.4

Review

RADICALS

Index

Radicand

When the index of the radical is not shown then it is understood to be an index of 2

Radical

𝟑√𝟔𝟒

=

EXAMPLE 1:

a)Give 4 examples of radicals

b)Use a different radicand and index for each radical

c) Explain the meaning of the index of each radical

Evaluate each radical:

√36

= 0.5

= 6= 2=

= 5

EXAMPLE 2:

Choose values of n and x so that is:

a) A whole number

b) A negative integer

c) A rational number

d) An approximate decimal

= 4

= 5/4

= 1.4141…

= -3

EXAMPLE 3:

4.2 Irrational Numbers

WORK WITH YOUR PARTNER

1. How are radicals that are rational numbers different from radicals that are not rational numbers?

Rational Numbers: Q a

b| a,b I ,b 0

These are rational numbers: These are NOT rational numbers:

2. Which of these radicals are rational numbers? Which ones are not rational numbers?

How do you know?

WORK WITH YOUR PARTNER

RATIONAL NUMBERSa. Can be written in the formb. Radicals that are square roots of perfect squares,

cube roots of perfect cubes etc..c. They have decimal representation which

terminate or repeats

Q a

b| a,b I ,b 0

IRATIONAL NUMBERS

a. Can not be written in the formb. They are non-repeating and non-terminating

decimals

Q a

b| a,b I ,b 0

EXAMPLE 1: Tell whether each number is rational or irrational. Explain how do you

know.

Rational, because 8/27 is a perfect cube. Also, 2/3 or 0.666… is a repeating decimal.

Irrational, because 14 is not a perfect square. Also, √14 is NOT a repeating decimal and DOES NOT

terminate

Rational, because 0.5 terminates.

Irrational, because π is not a repeating decimal and does not terminates

POWER POINT PRACTICE PROBLEMTell whether each number is rational or

irrational. Explain how do you know.

EXAMPLE 2:Use a number line to order these numbers from

least to greatest

Use Calculators!

-2 -1 0 1 2 3 4 5

EXAMPLE 2:Use a number line to order these numbers from

least to greatest

POWERPOINT PRACTICE PROBLEMUse a number line to order these numbers from

least to greatest

HOMEWORKO PAGES: 211 - 212O PROBLEMS: 3 – 6, 9, 15, 20, 18, 19

4.2

4.3 Mixed and Entire Radicals

Index

Radicand

Review of Radicals

When the index of the radical is not shown then it isunderstood to be an index of 2.

Radical

𝟑√𝟔𝟒 =

MULTIPLICATION PROPERTY of RADICALS

Use Your Calculator to calculate:

What do you notice?

𝒏√𝒂𝒃=𝒏√𝒂 ·𝒏√𝒃

WE USE THIS PROPERTY TO: Simplify square roots and cube roots

that are not perfect squares or perfect cubes, but have factors that are perfect squares/cubes

MULTIPLICATION PROPERTY of RADICALS

where n is a natural number, and a and b are real numbers

Example 1:❑√𝟐𝟒=√𝟒 ·√𝟔

¿𝟐 ·√𝟔¿𝟐√𝟔

Example 2:

𝟑√𝟐𝟒=𝟑√𝟑 ·𝟖¿𝟑√𝟑·𝟑√𝟖¿𝟐𝟑√𝟑

Simplify each radical.

Write each radical as a product of prime factors, then simplify.

Since √80 is a square root. Look for factors that appear twice

Simplify each radical.

Write each radical as a product of prime factors, then simplify.

Since 144 ∛ is a cube root. Look for factors that appear three times

Simplify each radical.

Write each radical as a product of prime factors, then simplify.

Since 162 ∜ is a fourth root. Look for factors that appear four times

POWERPOINT PRACTICE PROBLEMSimplify each radical.

Some numbers such as 200 have more than one perfect square factor:

For example, the factors of 200 are:1, 2 ,4, 5, 8, 10, 20, 25, 40, 50, 100,

200Since 1, 4, 16, 25, 100, and 400 are perfect squares, we can simplify √400 in several ways:

❑√𝟐𝟎𝟎=√𝟏 ·𝟐𝟎𝟎=𝟏√𝟐𝟎𝟎❑√𝟐𝟎𝟎=√𝟐𝟓 ·𝟖=√𝟐𝟓 ·√𝟖=𝟓√𝟖❑√𝟐𝟎𝟎=√𝟏𝟎𝟎 ·𝟐=√𝟏𝟎𝟎 ·√𝟐=𝟏𝟎 √𝟐

Writing Radicals in Simplest Form

Writing Radicals in Simplest Form

10√2 is in simplest form because the radical contains no perfect

square factors other than 1

❑√𝟐𝟎𝟎=√𝟏 ·𝟐𝟎𝟎=𝟏√𝟐𝟎𝟎❑√𝟐𝟎𝟎=√𝟐𝟓 ·𝟖=√𝟐𝟓 ·√𝟖=𝟓√𝟖❑√𝟐𝟎𝟎=√𝟏𝟎𝟎 ·𝟐=√𝟏𝟎𝟎 ·√𝟐=𝟏𝟎 √𝟐

Mixed Radical: the product of a number and a

radical

4 6Entire Radical:

the product of one and a radical

72

Writing Mixed Radicals as Entire RadicalsAny number can be written as the square

root of its square!

2 = 45 = 100 =

Any number can be also written as the cube root of its cube, or the fourth root of

its perfect fourth!2 =

45 =

𝒏√𝒂𝒃=𝒏√𝒂 ·𝒏√𝒃Writing Mixed Radicals as Entire

Radicals

𝒏√𝒂 ·𝒏√𝒃=𝒏√𝒂𝒃

Write each mixed radical as an entire radical

𝒏√𝒂 ·𝒏√𝒃=𝒏√𝒂𝒃

POWERPOINT PRACTICE PROBLEMWrite each mixed radical as an entire

radical

HOMEWORKO PAGES: 218 - 219O PROBLEMS: 4, 5, 7, 11 – 12 (a, d, e,

h, i), 15 – 18, 19, 20

4.3