Roots and Powers

Hughes, Cynthia, Cherie, Scarlett

Root & Power

Revie

w Time

1234

Irrational Number

Mixed and Entire Radicals

Fractional Exponents & Radicals

Negative Exponents,

Reciprocals,

and the Exponent Laws

Irrational NumberPart 1

By hughes

Classifying and Ordering Numbers

IRRATIONAL NUMBERS

Rational numbers are numbers that can be written in the form of a fraction or ratio, or more specifically as a quotient of integers

Any number that cannot be written as a quotient of integers is called an irrational number

∏ is one example of an irrational number….– √0.24, 3√9, √2, √1/3, 4√12, e

• Some examples of rational numbers?– √100, √0.25, 3√8, 0.5, 5/6, 7, 5√-32

Can you think of any more?

Rational numbers: Irrational numbers:

Rational Vs. Irrational

Numbers

You should have noticed that the decimal representation of a rational number either terminates, or repeats–0.5, 1.25, 3.675–1.3333…., 2.14141414…..

The decimal representation of an irrational number neither terminates nor repeats–3.14159265358………..

Which of these numbers are rational numbers and which are irrational numbers?

√1.44, √64/81, 3√-27, √4/5, √5

So……………………………

…………………….

R R R I I

Exact Values Vs. Approximate

Values

When an irrational number is written as a radical, for example; √2 or 3√-50, we say the radical is the exact value of the irrational number.

When we use a calculator to find the decimal value, we say this is an approximate value

We can approximate the location of an irrational number on a number line

Summary Of Number Sets

Example 1

• 3√13 ≈ 2.3513… • √18 ≈ 4.2426… • √9 = 3 • 4√27 ≈ 2.2795… • 3√-5 ≈ -1.7099…• From least to greatest: 3√-5, 4√27, 3√13, √9, √18

Order these numbers on a number line from least to greatest: 3√13, √18, √9, 4√27, 3√-5

Bones

Mixed and Entire

Radicals

Part 2

By Cherie

Example 1

Simplifying Radicals Using Prime Factorization

Simplify the radical √80

Solution:√80 = √8*10 = √2*2*2*5*2 = √(2*2)*(2*2)*5 = √4*√4*√5 =2*2*√5 =√5

Multiple AnswersSome numbers, such as 200, have more than

one perfect square factorThe factors of 200 are:

1,2,4,5,8,10,20,25,40,50,100,200Since 4, 25, and 100 are perfect squares, we

can simplify √200 in three ways:2√50, 5√8, 10√210√2 is in simplest form because the radical

contains no perfect square factors other than 1.

Example 2) Writing Radicals in Simplest Form

Write the radical in simplest form, if possible.3√40

Solution:Look for the perfect nth factors, where n is the index of the

radical.The factors of 40 are: 1,2,4,5,8,10,20,40The greatest perfect cube is 8 = 2*2*2, so write 40 as 8*5.3√40 = 3√8*5 = 3√8*3√5 = 23√5

Your turn:Write the radical in simplest form, if possible.

√26, 4√32Cannot be simplified, 24√2

Mixed and Entire Radicals

Radicals of the form n√x such as √80, or 3√144 are entire radicals

Radicals of the form an√x such as 4√5, or 23√18 are mixed radicals

(mixed radical entire radical)

Example 3) Writing Mixed Radicals as Entire Radicals

Write the mixed radical as an entire radical33√2

Solution:Write 3 as: 3√3*3*3 = 3√2733√2 = 3√27 * 3√2 = 3√27*2 = 3√54

Your turn:Write each mixed radical as an entire radical.4√3, 25√2

√48, 5√64

Review

Multiplication Property of Radicals is:n√ab = n√a * n√b, where n is a natural number, and a

and b are real numbersto write a radical of index n in simplest form, we

write the radicand as a product of 2 factors, one of which is the greatest perfect nth power

Radicals of the form n√x such as √80, or 3√144 are entire radicals

Radicals of the form an√x such as 4√5, or 23√18 are mixed radicals

Fractional Exponents and Radicals

Part 3

By Cynthia

Powers with rational

exponents with numerators 1

When n is a natural number and x is a rational number!!!!!!

Examples

Try it Evaluate each power without using a

calculator:271/3

271/3 = 3√27 = 3 0.491/2

0.491/2 = √0.49= 0.7

Powers with Rational

ExponentsWhen m and n are natural numbers, and x is a rational number:

xm/n = (x1/n)m = (n√x)m

And:xm/n = (xm)1/n = n√xm

TRY IT Write 402/3 in radical form in 2 ways

ANSWER:

Use am/n = (n√a)m or n√am

402/3 = (3√40)2 or 3√402

ReviewPowers with Rational Exponents with Numerator 1When n is a natural number and x is a rational number: x1/n = n√xPowers with rational exponentsWhen m and n are natural numbers, and x is a rational number:

xm/n = (x1/n)m = (n√x)m

And:xm/n = (xm)1/n = n√xm

Negative Exponents, Reciprocals,

and the Exponent Laws

Part 4

By Scarlett

Basic

Any two numbers that have a product of 1 are called reciprocals

Reciprocals

am x an = am+n

4 x ¼ = 12/3 x 3/2 = 1Also Applies to Powers

Basic

When x is any non-zero number and n is a rational number, x-n is the reciprocal of xn

x-n = (1/x)n (1/x)-n = xn, x ≠ 0

Laws of Exponents

Product of Powers: am x an = am+n

Quotient of Powers: am/an = am-n, a ≠ 0Power of a Power: (am)n = amn

Power of a Product: (ab)m = ambm

Power of a Quotient: (a/b)m = am/bm, b ≠ 0

Example

Example 1

Evaluate the power below:3-2

Solution:According to the Law: x-n = (1/x)n 3-2 = (1/3)2 1/9

Example

Example 2

Evaluate the power below:

(-3/4) -3

Solution:According to the Law:(1/x)-n = xn, x ≠ 0(-3/4)-3 = (-4/3)3

-64/27

Example

Example 3

Evaluate the power below:

8-2/3

Solution:8-2/3 = (1/8)2/3 =

1/(3√8)2

(1/2)2

¼ x-n = (1/x)n (1/x)-n = xn, x ≠ 0

Example

Example 4

Simplify the expression 4a-2b2/3/2a2b1/3

First use the quotient of powers law4/2 x a-2/a2 x b2/3/b1/3 = 2 x a(-2)-2

x b2/3-1/3

2a-4b1/3

Then write with a positive exponent2b1/3/a4

Your TurnEvaluate the power

below:(Choose only 2 of

them)

1) (9/16)-3/2

2) (7/24) -1/9

3) (9/20) 7/4

4) (25/10) -1/3

Made by Math Group#6

Scarlett, Cherie, Cynthia and

HughesThank You