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YOU are better than YOU think. Show
yourself how:
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Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
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
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
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.
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.
| |
More Algebra Hints
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.
Irreducible Case - Leave as is
If h is prime or a product of primes to the first power then no
simplification of the square root
__
Ö h
is possible. Leave as is.
Reducible Case if h is a perfect square
If h = n2 and n > 0 then
Examples
__
Ö 9 |
= 3 |
|
___
Ö 25 |
= 5 |
|
____
Ö 169 |
= 13 |
|
Combined Case
h = a2b is a product of a perfect square a and another number b
which may or not be irreducible.
For a > 0 and b > 0,
__
Ö h |
= |
___
Ö a2b |
= |
a |
__
Öb |
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 |
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 |
|
More Simplifications:
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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Vec & Cmplx No Applet
B2 C. Conjugates
B3 Pythagoras
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Simplification
Using formulas forwards
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Skills for Algebra!
What
is a Variable?
|