Processing Math: Done
Solution 2.1:8c
From Förberedande kurs i matematik 1
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- | {{ | + | When we come across large and complicated expressions, we have to work step by step; |
- | < | + | |
- | {{ | + | as a first goal, we can multiply the top and bottom of the fraction |
- | {{ | + | |
- | < | + | |
- | {{ | + | <math>\frac{1}{1+\frac{1}{1+x}}</math> |
+ | |||
+ | |||
+ | by | ||
+ | <math>1+x</math>, so as to reduce it to an expression having one fraction sign: | ||
+ | |||
+ | |||
+ | <math>\begin{align} | ||
+ | & \frac{1}{1+\frac{1}{1+\frac{1}{1+x}}}=\frac{1}{1+\frac{1}{1+\frac{1}{1+x}}\centerdot \frac{1+x}{1+x}}=\frac{1}{1+\frac{1+x}{\left( 1+\frac{1}{1+x} \right)\left( 1+x \right)}} \\ | ||
+ | & \\ | ||
+ | & =\frac{1}{1+\frac{1+x}{1+x+\frac{1+x}{1+x}}}=\frac{1}{1+\frac{1+x}{1+x+1}}=\frac{1}{1+\frac{x+1}{x+2}} \\ | ||
+ | \end{align}</math> | ||
+ | |||
+ | |||
+ | The next step is to multiply the top and bottom of our new expression by | ||
+ | <math>x+2</math>, | ||
+ | so as to obtain the final answer, | ||
+ | |||
+ | |||
+ | <math>\begin{align} | ||
+ | & \frac{1}{1+\frac{x+1}{x+2}}\centerdot \frac{x+2}{x+2}=\frac{x+2}{\left( 1+\frac{x+1}{x+2} \right)\left( x+2 \right)}=\frac{x+2}{x+2+\frac{x+1}{x+2}\left( x+2 \right)} \\ | ||
+ | & \\ | ||
+ | & \frac{x+2}{x+2+x+1}=\frac{x+2}{2x+3} \\ | ||
+ | & \\ | ||
+ | \end{align}</math> |
Revision as of 13:52, 16 September 2008
When we come across large and complicated expressions, we have to work step by step;
as a first goal, we can multiply the top and bottom of the fraction
by
1+x1+x=11+1+x
1+11+x
1+x
=11+1+x1+x+1+x1+x=11+1+x1+x+1=11+x+2x+1
The next step is to multiply the top and bottom of our new expression by
x+2x+2=x+2
1+x+2x+1
x+2
=x+2x+2+x+2x+1
x+2
x+2x+2+x+1=x+22x+3