Solve 1/frac{3{4}sqrt{2}+3/2}

$$\frac { 1 } { \frac { 3 } { 4 } \sqrt { 2 } + \frac { 3 } { 2 } }$$

Evaluate

$$\frac{4-2\sqrt{2}}{3}\approx 0.390524292$$

Solution Steps

Rationalize the denominator of \frac{1}{\frac{3}{4}\sqrt{2}+\frac{3}{2}} by multiplying numerator and denominator by \frac{3}{4}\sqrt{2}-\frac{3}{2}.

$$\frac{\frac{3}{4}\sqrt{2}-\frac{3}{2}}{\left(\frac{3}{4}\sqrt{2}+\frac{3}{2}\right)\left(\frac{3}{4}\sqrt{2}-\frac{3}{2}\right)}$$

Consider \left(\frac{3}{4}\sqrt{2}+\frac{3}{2}\right)\left(\frac{3}{4}\sqrt{2}-\frac{3}{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}.

$$\frac{\frac{3}{4}\sqrt{2}-\frac{3}{2}}{\left(\frac{3}{4}\sqrt{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}}$$

Expand \left(\frac{3}{4}\sqrt{2}\right)^{2}.

$$\frac{\frac{3}{4}\sqrt{2}-\frac{3}{2}}{\left(\frac{3}{4}\right)^{2}\left(\sqrt{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}}$$

Calculate \frac{3}{4} to the power of 2 and get \frac{9}{16}.

$$\frac{\frac{3}{4}\sqrt{2}-\frac{3}{2}}{\frac{9}{16}\left(\sqrt{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{3}{4}\sqrt{2}-\frac{3}{2}}{\frac{9}{16}\times 2-\left(\frac{3}{2}\right)^{2}}$$

Express \frac{9}{16}\times 2 as a single fraction.

$$\frac{\frac{3}{4}\sqrt{2}-\frac{3}{2}}{\frac{9\times 2}{16}-\left(\frac{3}{2}\right)^{2}}$$

Multiply 9 and 2 to get 18.

$$\frac{\frac{3}{4}\sqrt{2}-\frac{3}{2}}{\frac{18}{16}-\left(\frac{3}{2}\right)^{2}}$$

Reduce the fraction \frac{18}{16} to lowest terms by extracting and canceling out 2.

$$\frac{\frac{3}{4}\sqrt{2}-\frac{3}{2}}{\frac{9}{8}-\left(\frac{3}{2}\right)^{2}}$$

Calculate \frac{3}{2} to the power of 2 and get \frac{9}{4}.

$$\frac{\frac{3}{4}\sqrt{2}-\frac{3}{2}}{\frac{9}{8}-\frac{9}{4}}$$

Least common multiple of 8 and 4 is 8. Convert \frac{9}{8} and \frac{9}{4} to fractions with denominator 8.

$$\frac{\frac{3}{4}\sqrt{2}-\frac{3}{2}}{\frac{9}{8}-\frac{18}{8}}$$

Since \frac{9}{8} and \frac{18}{8} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3}{4}\sqrt{2}-\frac{3}{2}}{\frac{9-18}{8}}$$

Subtract 18 from 9 to get -9.

$$\frac{\frac{3}{4}\sqrt{2}-\frac{3}{2}}{-\frac{9}{8}}$$

Divide \frac{3}{4}\sqrt{2}-\frac{3}{2} by -\frac{9}{8} by multiplying \frac{3}{4}\sqrt{2}-\frac{3}{2} by the reciprocal of -\frac{9}{8}.

$$\frac{\left(\frac{3}{4}\sqrt{2}-\frac{3}{2}\right)\times 8}{-9}$$

Use the distributive property to multiply \frac{3}{4}\sqrt{2}-\frac{3}{2} by 8.

$$\frac{\frac{3}{4}\sqrt{2}\times 8-\frac{3}{2}\times 8}{-9}$$

Express \frac{3}{4}\times 8 as a single fraction.

$$\frac{\frac{3\times 8}{4}\sqrt{2}-\frac{3}{2}\times 8}{-9}$$

Multiply 3 and 8 to get 24.

$$\frac{\frac{24}{4}\sqrt{2}-\frac{3}{2}\times 8}{-9}$$

Divide 24 by 4 to get 6.

$$\frac{6\sqrt{2}-\frac{3}{2}\times 8}{-9}$$

Express -\frac{3}{2}\times 8 as a single fraction.

$$\frac{6\sqrt{2}+\frac{-3\times 8}{2}}{-9}$$

Multiply -3 and 8 to get -24.

$$\frac{6\sqrt{2}+\frac{-24}{2}}{-9}$$

Divide -24 by 2 to get -12.

$$\frac{6\sqrt{2}-12}{-9}$$

Factor

$$\frac{2 \sqrt{2} {(\sqrt{2} - 1)}}{3} = 0.39052429175127007$$