Solve 1/4/3+7+1/7-4sqrt{3}

$$\frac { 1 } { 4 / 3 + 7 } + \frac { 1 } { 7 - 4 \sqrt { 3 } }$$

Evaluate

$$4\sqrt{3}+\frac{178}{25}\approx 14.04820323$$

Solution Steps

Convert 7 to fraction \frac{21}{3}.

$$\frac{1}{\frac{4}{3}+\frac{21}{3}}+\frac{1}{7-4\sqrt{3}}$$

Since \frac{4}{3} and \frac{21}{3} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{4+21}{3}}+\frac{1}{7-4\sqrt{3}}$$

Add 4 and 21 to get 25.

$$\frac{1}{\frac{25}{3}}+\frac{1}{7-4\sqrt{3}}$$

Divide 1 by \frac{25}{3} by multiplying 1 by the reciprocal of \frac{25}{3}.

$$1\times \frac{3}{25}+\frac{1}{7-4\sqrt{3}}$$

Multiply 1 and \frac{3}{25} to get \frac{3}{25}.

$$\frac{3}{25}+\frac{1}{7-4\sqrt{3}}$$

Rationalize the denominator of \frac{1}{7-4\sqrt{3}} by multiplying numerator and denominator by 7+4\sqrt{3}.

$$\frac{3}{25}+\frac{7+4\sqrt{3}}{\left(7-4\sqrt{3}\right)\left(7+4\sqrt{3}\right)}$$

Consider \left(7-4\sqrt{3}\right)\left(7+4\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}.

$$\frac{3}{25}+\frac{7+4\sqrt{3}}{7^{2}-\left(-4\sqrt{3}\right)^{2}}$$

Calculate 7 to the power of 2 and get 49.

$$\frac{3}{25}+\frac{7+4\sqrt{3}}{49-\left(-4\sqrt{3}\right)^{2}}$$

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

$$\frac{3}{25}+\frac{7+4\sqrt{3}}{49-\left(-4\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -4 to the power of 2 and get 16.

$$\frac{3}{25}+\frac{7+4\sqrt{3}}{49-16\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{3}{25}+\frac{7+4\sqrt{3}}{49-16\times 3}$$

Multiply 16 and 3 to get 48.

$$\frac{3}{25}+\frac{7+4\sqrt{3}}{49-48}$$

Subtract 48 from 49 to get 1.

$$\frac{3}{25}+\frac{7+4\sqrt{3}}{1}$$

Anything divided by one gives itself.

$$\frac{3}{25}+7+4\sqrt{3}$$

Convert 7 to fraction \frac{175}{25}.

$$\frac{3}{25}+\frac{175}{25}+4\sqrt{3}$$

Since \frac{3}{25} and \frac{175}{25} have the same denominator, add them by adding their numerators.

$$\frac{3+175}{25}+4\sqrt{3}$$

Add 3 and 175 to get 178.

$$\frac{178}{25}+4\sqrt{3}$$