Evaluate
\frac{3\sqrt{6}+3\sqrt{10}}{2}\approx 8.417651104
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\frac{3\sqrt{2}\left(\sqrt{5}+\sqrt{3}\right)}{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}
Rationalize the denominator of \frac{3\sqrt{2}}{\sqrt{5}-\sqrt{3}} by multiplying numerator and denominator by \sqrt{5}+\sqrt{3}.
\frac{3\sqrt{2}\left(\sqrt{5}+\sqrt{3}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}
Consider \left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\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{3\sqrt{2}\left(\sqrt{5}+\sqrt{3}\right)}{5-3}
Square \sqrt{5}. Square \sqrt{3}.
\frac{3\sqrt{2}\left(\sqrt{5}+\sqrt{3}\right)}{2}
Subtract 3 from 5 to get 2.
\frac{3\sqrt{2}\sqrt{5}+3\sqrt{2}\sqrt{3}}{2}
Use the distributive property to multiply 3\sqrt{2} by \sqrt{5}+\sqrt{3}.
\frac{3\sqrt{10}+3\sqrt{2}\sqrt{3}}{2}
To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.
\frac{3\sqrt{10}+3\sqrt{6}}{2}
To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
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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}