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

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

Evaluate

$$\frac{10\sqrt{6}}{207}+45\approx 45.118332838$$

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}.

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

Multiply 69 and 3 to get 207.

$$45+12\times \frac{5}{207\sqrt{6}}$$

Rationalize the denominator of \frac{5}{207\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$45+12\times \frac{5\sqrt{6}}{207\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$45+12\times \frac{5\sqrt{6}}{207\times 6}$$

Multiply 207 and 6 to get 1242.

$$45+12\times \frac{5\sqrt{6}}{1242}$$

Express 12\times \frac{5\sqrt{6}}{1242} as a single fraction.

$$45+\frac{12\times 5\sqrt{6}}{1242}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 45 times \frac{1242}{1242}.

$$\frac{45\times 1242}{1242}+\frac{12\times 5\sqrt{6}}{1242}$$

Since \frac{45\times 1242}{1242} and \frac{12\times 5\sqrt{6}}{1242} have the same denominator, add them by adding their numerators.

$$\frac{45\times 1242+12\times 5\sqrt{6}}{1242}$$

Do the multiplications in 45\times 1242+12\times 5\sqrt{6}.

$$\frac{55890+60\sqrt{6}}{1242}$$

Factor

$$\frac{5 {(2 \sqrt{6} + 1863)}}{207} = 45.118332837815615$$