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Employ an online or offline tutor at your own risk from
AU:
tutorfinder.com.au
CDN :
findatutor.ca
CDN: .i-tutor.ca
CDN: Montreal
Tutors
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UK:
tutorhunt.com
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USA: wiziq.com
USA: ziizoo.com
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YOU are better than YOU think. Show
yourself how:
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Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful,
Edifying, Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens
eyes. Leads to greater precision.
in reading and writing
Do not leave here without it - Logic
mastery will develops critical thinking, improve reading and
writing, and give a firmer base for work and studies at many levels.
Good luck.
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Caution: Site advice is
approximately correct, for some circumstances, not all. Site How-TOs
are logically developed, but not tried and tested. That leaves
room for thought and refinement.. |
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After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site
area on solving
linear2007 Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
For online automated help in senior
high school maths & calculus, visit quickmath.com
For Automatic Calculus and Algebra Help with derivatives, integrals,
graphs, linear equations, matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different
range of services, some free, some not, all based on webmathematica.
Good luck.
|
Explore collaborative whiteboards from groupboard,
twiddla or
scriblink.
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The simplification is shown with or without the use of primes. Here
computation may equal not the decimal approximation but the algebraic or
cosmetic simplification of square roots. The examples below show how
factorization and prime decomposition, together or not, may be used in the
simplification process and also providing a stopping rule.
Real Player Videos give more examples. View
them before, after besides the text below.
Square Roots of Whole Numbers without a calculator
If you have a calculator, you may compute or represent the square root of a
number exactly or approximately. But in algebraic calculations (or shorthand
mathematical reasoning with letters and symbols), approximations are to be
avoided. The latter may be done using the following methods. Some of these
methods are cosmetic. But their use leads to a common or standard form for
expressions involving square roots.
If h is prime then no simplification of the square root
__
Ö h
is possible.
First Simplification:
If h = n2 and n > 0 then
Examples
__
Ö 9 |
= 3 |
|
___
Ö 25 |
= 5 |
|
____
Ö 169 |
= 13 |
|
Second Simplification:
For a > 0 and b > 0,
Examples
____
Ö 500 |
= |
______
Ö (100)5 |
= 10 |
__
Ö 5 |
____
Ö 27 |
= |
______
Ö 323 |
= 3 |
__
Ö 3 |
_____
Ö 1200 |
= |
_______
Ö (100)12 |
= 10 |
___
Ö 12 |
___
But Ö 12 |
= |
____
Ö 223 |
= 2 |
__
Ö 3
|
Therefore
_____
Ö 1200 |
= 10 |
__
Ö12 |
= 10(2 |
__
Ö 3 ) |
= 20 |
__
Ö 3 |
Second Simplification Revisited
If h = a2b where the prime factorization of b only includes
primes, but no powers of primes (other than 1). Then
Example
h= 1500 = 500*3 = 3*22*53 = = 3*22*52*5=
(22*52) 3*5 = (2*5)23*5
gives
____
Ö1500 |
= |
2*5 |
___
Ö3*5 |
= |
10 |
__
Ö15 |
|
Third Simplification:
For a > 0, b > 0 and c > 0,
_____
Ö a2b2c |
= |
ab |
__
Ö c |
Example
_____
Ö 1200 |
= 10 |
_______
Ö100*4*3 |
= (10*2) |
__
Ö 3 ) |
= 20 |
__
Ö 3 |
Suppose h = a2b where the prime factorization of b only includes
primes, but no powers of primes (other than 1).
- [Play
Video] 5 minutes - Calculation of Squares and Square Roots
for Natural Numbers without and with decimal approximations. Exact
representation of square roots without approximation requires not
using a calculating. That is important in algebra - the statement
and derivation of formulas.
- [Play
Video] 1¾ minutes - How to Compute Square Roots by
Factorization
- [Play
Video] 3 minutes - Computational Properties - More on square
computation by factorization.
- [Play
Video] 3 minutes - Examples of square root computation by
factorization.
- [Play
Video]3¾ minutes - Examples of square root computation
by prime factorization.
In algebra, this simplification rewrites square roots in a standard
form, a standard that may lead to a common representation of square
roots of whole numbers when they appear in formulas and the derivation
or justification of formulas.
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www.whyslopes.com
Number Theory
[ Back ] [ Up ] [ Next ]
Start of Number Theory
Origins of Counting or Tallying
Adding Wholes
Multipling Wholes
Distributive Law Preamble
Distributive Law for Wholes
Consequences
More Consequences
What is a Fraction
Compound Fractions
Number Theory
Continued
Decimal Place Value
Place Value Reinforcement
Comparison Method
Addition Method
Subtraction Methods
Multiplication Methods
Division Methods
Remainder Arithmetic I
Primes & Composites
Primes Factorization Theorem
Primes & Composites
Prime Factorization Aids
Prime Factorization Examples
Counting Whole No. Factors
Arithmetic Videos
Square Roots & Primes
Long Division Continued
Fractions & Decimals
Fractions as Decimals
1 = 0.999 Recurring
Infinite Decimals Expansion Arithmetic
Ratio of Simple Fractions
Ratio of Decimal Fractions
Unsigned Reals Numbers
Signed Coordinates
Plane Vectors
Horizontal Vectors
Adding Vector Multiplies
Adding Signed Numbers
Multiplying Signed Numbers
Distributive Law for Reals
Real Numbers Axioms
Remainder Arithmetic II
Related Site Pages:
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck
Food for thought: Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice..
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