YOU are better than YOU think. Show yourself how:
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Learn to read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention. Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence in
work |
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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
4 Square Root Simplification
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
| __ Ö h |
= n |
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 |
__ |
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
| ___ Ö a2b |
= | a | __ Öb |
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).
Videos
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. |
www.whyslopes.com
Algebra, Odds & Ends,
1. Hints for Exams 2A. Exact Arithmetic 2B. Fractions Briefly 3. What is a Variable? 4.. Square Roots 5. Straight Lines 6. Problem Solving Methods 7. Trig and Complex No. 8. Complex Applet 9. History of No.s 10. ln(x) and exp(x) 13. Rename the > Sign 14. Problems: Quadratics 15. Problems: Algebra Test 16. Problems: Linear Eqns I 17. Problems: Linear Eqns II 18. Problem Solving Hints 19. Functions & Sets 20. Independent Variables 21. Why Logic 22. Why Math 23. The 15 Times Table 24. The 20 Times Table 25. Algebra Formulas 26. On Learning Maths 27. Maths in Biology 28. Navigation +Time 29 Quibble-What is Algebra 30. Logic in Maths
Odd and Ends, Essays
Constant Retirement Rate Road Safety 3 Strikes Law in California. Math HOW-TOs 9 Steps in Maths
Study With Others: twiddla.com has developed a collaborative whiteboard with audio & text chat that allows students, tutors & teachers to explore & scribble on blank pages and copies of webpages together, If scribbling is awkward with one browser, try another.
In Volume 2, Three Skills for Algebra, Chapters 8 to 14 and postscript What is a Variable point to a greater & clear use of words in algebra. Chapter 14 introduces a 4th skill for algebra, an elaboration of the third: - The direct and indirect use of formulas, numerically and algebraically, is unifying theme that should be mentioned aloud, with words, in each and every use of formula.