Evaluate
-\frac{\sqrt{2}}{2}+1\approx 0.292893219
Factor
\frac{\sqrt{2} {(\sqrt{2} - 1)}}{2} = 0.29289321881345254
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\frac{2-\sqrt{2}}{\left(2+\sqrt{2}\right)\left(2-\sqrt{2}\right)}
Rationalize the denominator of \frac{1}{2+\sqrt{2}} by multiplying numerator and denominator by 2-\sqrt{2}.
\frac{2-\sqrt{2}}{2^{2}-\left(\sqrt{2}\right)^{2}}
Consider \left(2+\sqrt{2}\right)\left(2-\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
\frac{2-\sqrt{2}}{4-2}
Square 2. Square \sqrt{2}.
\frac{2-\sqrt{2}}{2}
Subtract 2 from 4 to get 2.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}