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\left(1-3\sqrt{2}\right)\left(\sqrt{2}+\frac{1}{\sqrt{2}}\right)\approx -6.878679656
Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2}\approx 4.242640687 as the product of square roots \sqrt{3^{2}}\sqrt{2}\approx 4.242640687. Take the square root of 3^{2}\approx 9.
\left(1-3\sqrt{2}\right)\left(\sqrt{2}+\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\right)\approx -6.878679656
Rationalize the denominator of \frac{1}{\sqrt{2}}\approx 0.707106781 by multiplying numerator and denominator by \sqrt{2}\approx 1.414213562.
\left(1-3\sqrt{2}\right)\left(\sqrt{2}+\frac{\sqrt{2}}{2}\right)\approx -6.878679656
The square of \sqrt{2}\approx 1.414213562 is 2.
\left(1-3\sqrt{2}\right)\times \left(\frac{3}{2}\right)\sqrt{2}\approx -6.878679656
Combine \sqrt{2}\approx 1.414213562 and \frac{\sqrt{2}}{2}\approx 0.707106781 to get \frac{3}{2}\sqrt{2}\approx 2.121320344.
\left(\frac{3}{2}-3\sqrt{2}\times \left(\frac{3}{2}\right)\right)\sqrt{2}\approx -6.878679656
Use the distributive property to multiply 1-3\sqrt{2}\approx -3.242640687 by \frac{3}{2}=1.5.
\left(\frac{3}{2}+\frac{-3\times 3}{2}\sqrt{2}\right)\sqrt{2}\approx -6.878679656
Express -3\times \left(\frac{3}{2}\right)=-4.5 as a single fraction.
\left(\frac{3}{2}+\frac{-9}{2}\sqrt{2}\right)\sqrt{2}\approx -6.878679656
Multiply -3 and 3 to get -9.
\left(\frac{3}{2}-\frac{9}{2}\sqrt{2}\right)\sqrt{2}\approx -6.878679656
Fraction \frac{-9}{2}=-4.5 can be rewritten as -\frac{9}{2}=-4.5 by extracting the negative sign.
\frac{3}{2}\sqrt{2}-\frac{9}{2}\sqrt{2}\sqrt{2}\approx -6.878679656
Use the distributive property to multiply \frac{3}{2}-\frac{9}{2}\sqrt{2}\approx -4.863961031 by \sqrt{2}\approx 1.414213562.
\frac{3}{2}\sqrt{2}-\frac{9}{2}\times 2\approx -6.878679656
Multiply \sqrt{2}\approx 1.414213562 and \sqrt{2}\approx 1.414213562 to get 2.
\frac{3}{2}\sqrt{2}-9\approx -6.878679656
Cancel out 2 and 2.