From Förberedande kurs i matematik 1

Revision as of 09:08, 9 September 2008 (edit) Tekbot (Talk | contribs) m (Lösning 3.1:2e moved to Solution 3.1:2e: Robot: moved page) ← Previous diff		Current revision (08:07, 30 September 2008) (edit) (undo) Tek (Talk | contribs) m
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-	{{NAVCONTENT_START}}	+	Looking first at <math>\sqrt{18}</math> this square root expression can be simplified by writing 18 as a product of its smallest possible integer factors
-	<center> [[Image:3_1_2e.gif]] </center>	+
-	{{NAVCONTENT_STOP}}	+	{{Displayed math||<math>18 = 2\cdot 9 = 2\cdot 3\cdot 3 = 2\cdot 3^{2}</math>}}
		+
		+	and then we can take the quadratic out of the square root sign by using the rule
		+	<math>\sqrt{a^{2}b}=a\sqrt{b}</math> (valid for non-negative ''a'' and ''b''),
		+
		+	{{Displayed math||<math>\sqrt{18} = \sqrt{2\cdot 3^{2}} = 3\sqrt{2}\,\textrm{.}</math>}}
		+
		+	In the same way, we write <math>8 = 2\cdot 4 = 2\cdot 2\cdot 2 = 2^{3}</math> and get
		+
		+	{{Displayed math||<math>\sqrt{8} = \sqrt{2\cdot 2^{2}} = 2\sqrt{2}\,\textrm{.}</math>}}
		+
		+	All together, we get
		+
		+	{{Displayed math||<math>\begin{align}
		+	\sqrt{18}\sqrt{8}
		+	&= 3\sqrt{2}\cdot 2\sqrt{2}\\[5pt]
		+	&= 3\cdot 2\cdot \bigl(\sqrt{2}\bigr)^{2}\\[5pt]
		+	&= 3\cdot 2\cdot 2\\[5pt]
		+	&= 12\,\textrm{.}
		+	\end{align}</math>}}

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Current revision

Looking first at \displaystyle \sqrt{18} this square root expression can be simplified by writing 18 as a product of its smallest possible integer factors

\displaystyle 18 = 2\cdot 9 = 2\cdot 3\cdot 3 = 2\cdot 3^{2}

and then we can take the quadratic out of the square root sign by using the rule \displaystyle \sqrt{a^{2}b}=a\sqrt{b} (valid for non-negative a and b),

\displaystyle \sqrt{18} = \sqrt{2\cdot 3^{2}} = 3\sqrt{2}\,\textrm{.}

In the same way, we write \displaystyle 8 = 2\cdot 4 = 2\cdot 2\cdot 2 = 2^{3} and get

\displaystyle \sqrt{8} = \sqrt{2\cdot 2^{2}} = 2\sqrt{2}\,\textrm{.}

All together, we get

\displaystyle \begin{align} \sqrt{18}\sqrt{8} &= 3\sqrt{2}\cdot 2\sqrt{2}\\[5pt] &= 3\cdot 2\cdot \bigl(\sqrt{2}\bigr)^{2}\\[5pt] &= 3\cdot 2\cdot 2\\[5pt] &= 12\,\textrm{.} \end{align}