Solve 12/sqrt{7*3}*5/8-3sqrt{7}

$$\frac { 12 } { \sqrt { 7 } \times 3 } \cdot \frac { 5 } { 8 - 3 \sqrt { 7 } }$$

Evaluate

$$\frac{160\sqrt{7}}{7}+60\approx 120.474315681$$

Solution Steps

Rationalize the denominator of \frac{12}{3\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{12\sqrt{7}}{3\left(\sqrt{7}\right)^{2}}\times \frac{5}{8-3\sqrt{7}}$$

The square of \sqrt{7} is 7.

$$\frac{12\sqrt{7}}{3\times 7}\times \frac{5}{8-3\sqrt{7}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{4\sqrt{7}}{7}\times \frac{5}{8-3\sqrt{7}}$$

Rationalize the denominator of \frac{5}{8-3\sqrt{7}} by multiplying numerator and denominator by 8+3\sqrt{7}.

$$\frac{4\sqrt{7}}{7}\times \frac{5\left(8+3\sqrt{7}\right)}{\left(8-3\sqrt{7}\right)\left(8+3\sqrt{7}\right)}$$

Consider \left(8-3\sqrt{7}\right)\left(8+3\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{4\sqrt{7}}{7}\times \frac{5\left(8+3\sqrt{7}\right)}{8^{2}-\left(-3\sqrt{7}\right)^{2}}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{4\sqrt{7}}{7}\times \frac{5\left(8+3\sqrt{7}\right)}{64-\left(-3\sqrt{7}\right)^{2}}$$

Expand \left(-3\sqrt{7}\right)^{2}.

$$\frac{4\sqrt{7}}{7}\times \frac{5\left(8+3\sqrt{7}\right)}{64-\left(-3\right)^{2}\left(\sqrt{7}\right)^{2}}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{4\sqrt{7}}{7}\times \frac{5\left(8+3\sqrt{7}\right)}{64-9\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{4\sqrt{7}}{7}\times \frac{5\left(8+3\sqrt{7}\right)}{64-9\times 7}$$

Multiply 9 and 7 to get 63.

$$\frac{4\sqrt{7}}{7}\times \frac{5\left(8+3\sqrt{7}\right)}{64-63}$$

Subtract 63 from 64 to get 1.

$$\frac{4\sqrt{7}}{7}\times \frac{5\left(8+3\sqrt{7}\right)}{1}$$

Anything divided by one gives itself.

$$\frac{4\sqrt{7}}{7}\times 5\left(8+3\sqrt{7}\right)$$

Use the distributive property to multiply 5 by 8+3\sqrt{7}.

$$\frac{4\sqrt{7}}{7}\left(40+15\sqrt{7}\right)$$

Express \frac{4\sqrt{7}}{7}\left(40+15\sqrt{7}\right) as a single fraction.

$$\frac{4\sqrt{7}\left(40+15\sqrt{7}\right)}{7}$$

Use the distributive property to multiply 4\sqrt{7} by 40+15\sqrt{7}.

$$\frac{160\sqrt{7}+60\left(\sqrt{7}\right)^{2}}{7}$$

The square of \sqrt{7} is 7.

$$\frac{160\sqrt{7}+60\times 7}{7}$$

Multiply 60 and 7 to get 420.

$$\frac{160\sqrt{7}+420}{7}$$