Solve for P
P=5-2\sqrt{6}\approx 0.101020514
Assign P
P≔5-2\sqrt{6}
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P=\frac{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)}{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)}
Rationalize the denominator of \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}} by multiplying numerator and denominator by \sqrt{3}-\sqrt{2}.
P=\frac{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)}{\left(\sqrt{3}\right)^{2}-\left(\sqrt{2}\right)^{2}}
Consider \left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}-\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}.
P=\frac{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)}{3-2}
Square \sqrt{3}. Square \sqrt{2}.
P=\frac{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)}{1}
Subtract 2 from 3 to get 1.
P=\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)
Anything divided by one gives itself.
P=\left(\sqrt{3}-\sqrt{2}\right)^{2}
Multiply \sqrt{3}-\sqrt{2} and \sqrt{3}-\sqrt{2} to get \left(\sqrt{3}-\sqrt{2}\right)^{2}.
P=\left(\sqrt{3}\right)^{2}-2\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{3}-\sqrt{2}\right)^{2}.
P=3-2\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}
The square of \sqrt{3} is 3.
P=3-2\sqrt{6}+\left(\sqrt{2}\right)^{2}
To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.
P=3-2\sqrt{6}+2
The square of \sqrt{2} is 2.
P=5-2\sqrt{6}
Add 3 and 2 to get 5.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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y = 3x + 4
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Matrix
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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}