3.1 Roots - Revision history

Jdinglis at 19:32, 26 October 2008

Jdinglis — Sun, 26 Oct 2008 19:32:47 GMT

←Older revision		Revision as of 19:32, 26 October 2008
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	For those of you who want to deepen your understanding or need more detailed explanations consider the following references		For those of you who want to deepen your understanding or need more detailed explanations consider the following references

-	[http://en.wikipedia.org/wiki/Root_(mathematics) Learn more about square roots ~~in the English~~ Wikipedia ]	+	[http://en.wikipedia.org/wiki/Root_(mathematics) Learn more about square roots from Wikipedia ]

	[http://www.mathacademy.com/pr/prime/articles/irr2/ How do we know that the square root of 2 is not a fraction?]		[http://www.mathacademy.com/pr/prime/articles/irr2/ How do we know that the square root of 2 is not a fraction?]

Jdinglis at 14:01, 26 October 2008

Jdinglis — Sun, 26 Oct 2008 14:01:39 GMT

←Older revision		Revision as of 14:01, 26 October 2008
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	'''Study advice'''		'''Study advice'''

-	'''Basic and final tests:'''	+	'''Basic and final tests'''

	After you have read the text and worked through the exercises, you should do the basic and final tests to pass this section. You can find the link to the tests in your student lounge.		After you have read the text and worked through the exercises, you should do the basic and final tests to pass this section. You can find the link to the tests in your student lounge.

-	'''Keep in mind that: '''	+	'''Keep in mind that... '''

	The square root of a number is always non-negative (that is, positive or zero)!		The square root of a number is always non-negative (that is, positive or zero)!
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-	'''Reviews:'''	+	'''Reviews'''

	For those of you who want to deepen your understanding or need more detailed explanations consider the following references		For those of you who want to deepen your understanding or need more detailed explanations consider the following references
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-	'''Useful web sites:'''	+	'''Useful web sites'''

	[http://mathforum.org/dr.math/faq/faq.sqrt.by.hand.html How to find the root of a number, without the help of calculators?]		[http://mathforum.org/dr.math/faq/faq.sqrt.by.hand.html How to find the root of a number, without the help of calculators?]
	</div>		</div>

Jdinglis at 17:04, 24 October 2008

Jdinglis — Fri, 24 Oct 2008 17:04:24 GMT

←Older revision		Revision as of 17:04, 24 October 2008
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	[http://en.wikipedia.org/wiki/Root_(mathematics) Learn more about square roots in the English Wikipedia ]		[http://en.wikipedia.org/wiki/Root_(mathematics) Learn more about square roots in the English Wikipedia ]

-	[http://www.mathacademy.com/pr/prime/articles/irr2/ How do we know that the ~~<math>\sqrt{~~2~~}</math>~~ is not a fraction?]	+	[http://www.mathacademy.com/pr/prime/articles/irr2/ How do we know that the square root of 2 is not a fraction?]

Jdinglis at 17:03, 24 October 2008

Jdinglis — Fri, 24 Oct 2008 17:03:03 GMT

←Older revision		Revision as of 17:03, 24 October 2008
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	The square root of a number is always non-negative (that is, positive or zero)!		The square root of a number is always non-negative (that is, positive or zero)!

-	Rules for roots are actually a special case of laws of exponents, since, for example, <math>\sqrt{x} = x^{1/2}<\math>.	+	Rules for roots are actually a special case of laws of exponents, since, for example, <math>\sqrt{x} = x^{1/2}</math>.

Jdinglis at 17:02, 24 October 2008

Jdinglis — Fri, 24 Oct 2008 17:02:04 GMT

←Older revision		Revision as of 17:02, 24 October 2008
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	'''Study advice'''		'''Study advice'''

-	'''Basic and final tests'''	+	'''Basic and final tests:'''

	After you have read the text and worked through the exercises, you should do the basic and final tests to pass this section. You can find the link to the tests in your student lounge.		After you have read the text and worked through the exercises, you should do the basic and final tests to pass this section. You can find the link to the tests in your student lounge.
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	The square root of a number is always non-negative (that is, positive or zero)!		The square root of a number is always non-negative (that is, positive or zero)!

-	Rules for roots are actually a special case of laws of exponents .	+	Rules for roots are actually a special case of laws of exponents, since, for example, <math>\sqrt{x} = x^{1/2}<\math>.

-	For example: <math>\sqrt{x}=x^{1/2}</math>.

		+	'''Reviews:'''

-	~~'''Reviews'''~~	+	For those of you who want to deepen your understanding or need more detailed explanations consider the following references
-		+
-	For those of you who want to deepen your ~~studies~~ or need more detailed explanations consider the following references	+

	[http://en.wikipedia.org/wiki/Root_(mathematics) Learn more about square roots in the English Wikipedia ]		[http://en.wikipedia.org/wiki/Root_(mathematics) Learn more about square roots in the English Wikipedia ]

-	[http://www.mathacademy.com/pr/prime/articles/irr2/ How do we know that the ~~root of~~ 2 is not a fraction?]	+	[http://www.mathacademy.com/pr/prime/articles/irr2/ How do we know that the <math>\sqrt{2}</math> is not a fraction?]


-	'''Useful web sites'''	+	'''Useful web sites:'''

	[http://mathforum.org/dr.math/faq/faq.sqrt.by.hand.html How to find the root of a number, without the help of calculators?]		[http://mathforum.org/dr.math/faq/faq.sqrt.by.hand.html How to find the root of a number, without the help of calculators?]
	</div>		</div>

Jdinglis at 16:56, 24 October 2008

Jdinglis — Fri, 24 Oct 2008 16:56:01 GMT

←Older revision		Revision as of 16:56, 24 October 2008
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	which usually is preferable.		which usually is preferable.

-	In other cases, ~~you~~ can ~~take advantage of~~ the difference of two squares ~~method,~~ <math>(a+b)(a-b) = a^2 – b^2</math>. ~~One multiplies~~ the numerator and denominator by the denominator´s “conjugate” expression ~~and~~ the root ~~sign is eliminated from~~ the denominator ~~by squaring, as in the following~~,	+	In other cases, we can use the difference of two squares i.e <math>(a+b)(a-b) = a^2 – b^2</math> to eliminate the root from the denominator. Indeed, if we multiply both the numerator and denominator by the denominator´s “conjugate” expression, then the root in the denominator gets squared and hence is removed. For example,

	{{Displayed math\|\|<math>\begin{align*}		{{Displayed math\|\|<math>\begin{align*}

Jdinglis at 16:50, 24 October 2008

Jdinglis — Fri, 24 Oct 2008 16:50:32 GMT

←Older revision		Revision as of 16:50, 24 October 2008
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	== Rational root expressions ==		== Rational root expressions ==

-	When roots appear in a rational expression it is often useful to write the expression in a form which does not contain roots in the denominator. This is because it is difficult to divide by irrational numbers in calculations by hand. ~~By multiplying~~ the ~~numerator and denominator~~ by <math> \sqrt{2} </math> ~~for example~~, one obtains	+	When roots appear in a rational expression it is often useful to write the expression in a form which does not contain roots in the denominator. This is because it is difficult to divide by irrational numbers in calculations by hand. In the example below, multiplying by <math> 1=\frac{\sqrt{2}}{\sqrt{2}} </math>, one obtains

	{{Displayed math\|\|<math>\frac{1}{\sqrt{2}}		{{Displayed math\|\|<math>\frac{1}{\sqrt{2}}

Jdinglis at 16:46, 24 October 2008

Jdinglis — Fri, 24 Oct 2008 16:46:12 GMT

←Older revision		Revision as of 16:46, 24 October 2008
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	== Rational root expressions ==		== Rational root expressions ==

-	When roots appear in a rational expression ~~one~~ often ~~wants~~ to ~~avoid~~ roots in the denominator (because it is difficult ~~with hand calculations~~ to divide by irrational numbers). By multiplying the numerator and denominator by <math> \sqrt{2} </math> for example, one obtains	+	When roots appear in a rational expression it is often useful to write the expression in a form which does not contain roots in the denominator. This is because it is difficult to divide by irrational numbers in calculations by hand. By multiplying the numerator and denominator by <math> \sqrt{2} </math> for example, one obtains

	{{Displayed math\|\|<math>\frac{1}{\sqrt{2}}		{{Displayed math\|\|<math>\frac{1}{\sqrt{2}}

Jdinglis at 16:40, 24 October 2008

Jdinglis — Fri, 24 Oct 2008 16:40:55 GMT

←Older revision		Revision as of 16:40, 24 October 2008
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	\sqrt[\scriptstyle n]{b}\\[4pt]		\sqrt[\scriptstyle n]{b}\\[4pt]
	\sqrt[\scriptstyle n]{\frac{a}{b}}		\sqrt[\scriptstyle n]{\frac{a}{b}}
-	&= \frac{\sqrt[\scriptstyle n]{a}}{\sqrt[\scriptstyle n]{b}}\\[4pt]	+	&= \frac{\sqrt[\scriptstyle n]{a}}{\sqrt[\scriptstyle n]{b}}, \quad\ \mathrm{for}\ b\neq0\\[4pt]
	a\,\sqrt[\scriptstyle n]{b}		a\,\sqrt[\scriptstyle n]{b}
	&= \sqrt[\scriptstyle n]{a^nb}		&= \sqrt[\scriptstyle n]{a^nb}
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	== Simplification of expressions containing roots ==		== Simplification of expressions containing roots ==

-	Often, one can significantly simplify expressions containing roots by using the ~~usual methods for roots~~ . As is also the case when ~~using the laws of~~ exponents, it is desirable to reduce expressions into as "small" roots ~~as possible~~. For example, it is a good idea to do the following	+	Often one can significantly simplify expressions containing roots by using the rules described above. As is also the case when dealing with exponents, it is desirable to reduce expressions into "small" roots. For example, it is a good idea to do the following

	{{Displayed math\|\|<math>\sqrt{8}		{{Displayed math\|\|<math>\sqrt{8}

Jdinglis at 16:34, 24 October 2008

Jdinglis — Fri, 24 Oct 2008 16:34:09 GMT

←Older revision		Revision as of 16:34, 24 October 2008
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	Note that, unlike square roots, cube roots are also defined for negative numbers.		Note that, unlike square roots, cube roots are also defined for negative numbers.

-	For any positive ~~integers~~ <math>n</math> one can define the <math>n</math>'th root of a number <math>a</math> as	+	For any positive integer <math>n</math> one can define the <math>n</math>'th root of a number <math>a</math>:

	* if <math>n</math> is even and <math>a\ge0</math> then <math>\sqrt[\scriptstyle n]{a}</math> is the non-negative number that when multiplied by itself <math>n</math> times gives <math>a</math>,		* if <math>n</math> is even and <math>a\ge0</math> then <math>\sqrt[\scriptstyle n]{a}</math> is the non-negative number that when multiplied by itself <math>n</math> times gives <math>a</math>,