Roots and Powers
description
Transcript of Roots and Powers
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