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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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
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
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