Solve 1/(3sqrt{3/7frac{sqrt{91}}{10})-(-1/7*-3/10)}

$$\frac { 1 } { ( \frac { 3 \sqrt { 3 } } { 7 } \frac { \sqrt { 91 } } { 10 } ) - ( - \frac { 1 } { 7 } \cdot - \frac { 3 } { 10 } ) }$$

Evaluate

$$\frac{35\left(\sqrt{273}+1\right)}{408}\approx 1.503173793$$

Short Solution Steps

Multiply \frac{3\sqrt{3}}{7} times \frac{\sqrt{91}}{10} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{\frac{3\sqrt{3}\sqrt{91}}{7\times 10}-\left(-\frac{1}{7}\left(-\frac{3}{10}\right)\right)}$$

Multiply -\frac{1}{7} times -\frac{3}{10} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{\frac{3\sqrt{3}\sqrt{91}}{7\times 10}-\frac{-\left(-3\right)}{7\times 10}}$$

Do the multiplications in the fraction \frac{-\left(-3\right)}{7\times 10}.

$$\frac{1}{\frac{3\sqrt{3}\sqrt{91}}{7\times 10}-\frac{3}{70}}$$

To add or subtract expressions, expand them to make their denominators the same. Expand 7\times 10.

$$\frac{1}{\frac{3\sqrt{3}\sqrt{91}}{70}-\frac{3}{70}}$$

Since \frac{3\sqrt{3}\sqrt{91}}{70} and \frac{3}{70} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{\frac{3\sqrt{3}\sqrt{91}-3}{70}}$$

Do the multiplications in 3\sqrt{3}\sqrt{91}-3.

$$\frac{1}{\frac{3\sqrt{273}-3}{70}}$$

Divide 1 by \frac{3\sqrt{273}-3}{70} by multiplying 1 by the reciprocal of \frac{3\sqrt{273}-3}{70}.

$$\frac{70}{3\sqrt{273}-3}$$

Rationalize the denominator of \frac{70}{3\sqrt{273}-3} by multiplying numerator and denominator by 3\sqrt{273}+3.

$$\frac{70\left(3\sqrt{273}+3\right)}{\left(3\sqrt{273}-3\right)\left(3\sqrt{273}+3\right)}$$

Consider \left(3\sqrt{273}-3\right)\left(3\sqrt{273}+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}.

$$\frac{70\left(3\sqrt{273}+3\right)}{\left(3\sqrt{273}\right)^{2}-3^{2}}$$

Expand \left(3\sqrt{273}\right)^{2}.

$$\frac{70\left(3\sqrt{273}+3\right)}{3^{2}\left(\sqrt{273}\right)^{2}-3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{70\left(3\sqrt{273}+3\right)}{9\left(\sqrt{273}\right)^{2}-3^{2}}$$

The square of \sqrt{273} is 273.

$$\frac{70\left(3\sqrt{273}+3\right)}{9\times 273-3^{2}}$$

Multiply 9 and 273 to get 2457.

$$\frac{70\left(3\sqrt{273}+3\right)}{2457-3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{70\left(3\sqrt{273}+3\right)}{2457-9}$$

Subtract 9 from 2457 to get 2448.

$$\frac{70\left(3\sqrt{273}+3\right)}{2448}$$

Divide 70\left(3\sqrt{273}+3\right) by 2448 to get \frac{35}{1224}\left(3\sqrt{273}+3\right).

$$\frac{35}{1224}\left(3\sqrt{273}+3\right)$$

Use the distributive property to multiply \frac{35}{1224} by 3\sqrt{273}+3.

$$\frac{35}{1224}\times 3\sqrt{273}+\frac{35}{1224}\times 3$$

Express \frac{35}{1224}\times 3 as a single fraction.

$$\frac{35\times 3}{1224}\sqrt{273}+\frac{35}{1224}\times 3$$

Multiply 35 and 3 to get 105.

$$\frac{105}{1224}\sqrt{273}+\frac{35}{1224}\times 3$$

Reduce the fraction \frac{105}{1224} to lowest terms by extracting and canceling out 3.

$$\frac{35}{408}\sqrt{273}+\frac{35}{1224}\times 3$$

Express \frac{35}{1224}\times 3 as a single fraction.

$$\frac{35}{408}\sqrt{273}+\frac{35\times 3}{1224}$$

Multiply 35 and 3 to get 105.

$$\frac{35}{408}\sqrt{273}+\frac{105}{1224}$$

Reduce the fraction \frac{105}{1224} to lowest terms by extracting and canceling out 3.

$$\frac{35}{408}\sqrt{273}+\frac{35}{408}$$

Factor

$$\frac{35 {(\sqrt{273} + 1)}}{408} = 1.5031737928064721$$