Solve 45+12*5/69+sqrt{54}

$$45+12 \times \frac{ 5 }{ 69+ \sqrt{ 54 } }$$

Evaluate

$$\frac{23995-20\sqrt{6}}{523}\approx 45.785870373$$

Short Solution Steps

Factor 54=3^{2}\times 6. Rewrite the square root of the product \sqrt{3^{2}\times 6} as the product of square roots \sqrt{3^{2}}\sqrt{6}. Take the square root of 3^{2}.

$$45+12\times \frac{5}{69+3\sqrt{6}}$$

Rationalize the denominator of \frac{5}{69+3\sqrt{6}} by multiplying numerator and denominator by 69-3\sqrt{6}.

$$45+12\times \frac{5\left(69-3\sqrt{6}\right)}{\left(69+3\sqrt{6}\right)\left(69-3\sqrt{6}\right)}$$

Consider \left(69+3\sqrt{6}\right)\left(69-3\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$45+12\times \frac{5\left(69-3\sqrt{6}\right)}{69^{2}-\left(3\sqrt{6}\right)^{2}}$$

Calculate 69 to the power of 2 and get 4761.

$$45+12\times \frac{5\left(69-3\sqrt{6}\right)}{4761-\left(3\sqrt{6}\right)^{2}}$$

Expand \left(3\sqrt{6}\right)^{2}.

$$45+12\times \frac{5\left(69-3\sqrt{6}\right)}{4761-3^{2}\left(\sqrt{6}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$45+12\times \frac{5\left(69-3\sqrt{6}\right)}{4761-9\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$45+12\times \frac{5\left(69-3\sqrt{6}\right)}{4761-9\times 6}$$

Multiply 9 and 6 to get 54.

$$45+12\times \frac{5\left(69-3\sqrt{6}\right)}{4761-54}$$

Subtract 54 from 4761 to get 4707.

$$45+12\times \frac{5\left(69-3\sqrt{6}\right)}{4707}$$

Express 12\times \frac{5\left(69-3\sqrt{6}\right)}{4707} as a single fraction.

$$45+\frac{12\times 5\left(69-3\sqrt{6}\right)}{4707}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 45 times \frac{4707}{4707}.

$$\frac{45\times 4707}{4707}+\frac{12\times 5\left(69-3\sqrt{6}\right)}{4707}$$

Since \frac{45\times 4707}{4707} and \frac{12\times 5\left(69-3\sqrt{6}\right)}{4707} have the same denominator, add them by adding their numerators.

$$\frac{45\times 4707+12\times 5\left(69-3\sqrt{6}\right)}{4707}$$

Do the multiplications in 45\times 4707+12\times 5\left(69-3\sqrt{6}\right).

$$\frac{211815+4140-180\sqrt{6}}{4707}$$

Do the calculations in 211815+4140-180\sqrt{6}.

$$\frac{215955-180\sqrt{6}}{4707}$$

Factor

$$\frac{5 {(4799 - 4 \sqrt{6})}}{523} = 45.78587037312492$$