Evaluate
\frac{\sqrt{3}}{4}-\frac{\sqrt{15}}{20}\approx 0.239363535
Share
Copied to clipboard
\frac{\sqrt{3}\left(5-\sqrt{5}\right)}{\left(5+\sqrt{5}\right)\left(5-\sqrt{5}\right)}
Rationalize the denominator of \frac{\sqrt{3}}{5+\sqrt{5}} by multiplying numerator and denominator by 5-\sqrt{5}.
\frac{\sqrt{3}\left(5-\sqrt{5}\right)}{5^{2}-\left(\sqrt{5}\right)^{2}}
Consider \left(5+\sqrt{5}\right)\left(5-\sqrt{5}\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{\sqrt{3}\left(5-\sqrt{5}\right)}{25-5}
Square 5. Square \sqrt{5}.
\frac{\sqrt{3}\left(5-\sqrt{5}\right)}{20}
Subtract 5 from 25 to get 20.
\frac{5\sqrt{3}-\sqrt{3}\sqrt{5}}{20}
Use the distributive property to multiply \sqrt{3} by 5-\sqrt{5}.
\frac{5\sqrt{3}-\sqrt{15}}{20}
To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.
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}