Solve sqrt{425}-8/3+sin60

$$\sqrt { 425 } - \frac { 8 } { 3 + \sin 60 }$$

Evaluate

$$\frac{16\sqrt{3}}{33}+5\sqrt{17}-\frac{32}{11}\approx 18.546219429$$

Short Solution Steps

Factor 425=5^{2}\times 17. Rewrite the square root of the product \sqrt{5^{2}\times 17} as the product of square roots \sqrt{5^{2}}\sqrt{17}. Take the square root of 5^{2}.

$$5\sqrt{17}-\frac{8}{3+\sin(60)}$$

Get the value of \sin(60) from trigonometric values table.

$$5\sqrt{17}-\frac{8}{3+\frac{\sqrt{3}}{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{2}{2}.

$$5\sqrt{17}-\frac{8}{\frac{3\times 2}{2}+\frac{\sqrt{3}}{2}}$$

Since \frac{3\times 2}{2} and \frac{\sqrt{3}}{2} have the same denominator, add them by adding their numerators.

$$5\sqrt{17}-\frac{8}{\frac{3\times 2+\sqrt{3}}{2}}$$

Do the multiplications in 3\times 2+\sqrt{3}.

$$5\sqrt{17}-\frac{8}{\frac{6+\sqrt{3}}{2}}$$

Divide 8 by \frac{6+\sqrt{3}}{2} by multiplying 8 by the reciprocal of \frac{6+\sqrt{3}}{2}.

$$5\sqrt{17}-\frac{8\times 2}{6+\sqrt{3}}$$

Rationalize the denominator of \frac{8\times 2}{6+\sqrt{3}} by multiplying numerator and denominator by 6-\sqrt{3}.

$$5\sqrt{17}-\frac{8\times 2\left(6-\sqrt{3}\right)}{\left(6+\sqrt{3}\right)\left(6-\sqrt{3}\right)}$$

Consider \left(6+\sqrt{3}\right)\left(6-\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}.

$$5\sqrt{17}-\frac{8\times 2\left(6-\sqrt{3}\right)}{6^{2}-\left(\sqrt{3}\right)^{2}}$$

Square 6. Square \sqrt{3}.

$$5\sqrt{17}-\frac{8\times 2\left(6-\sqrt{3}\right)}{36-3}$$

Subtract 3 from 36 to get 33.

$$5\sqrt{17}-\frac{8\times 2\left(6-\sqrt{3}\right)}{33}$$

Multiply 8 and 2 to get 16.

$$5\sqrt{17}-\frac{16\left(6-\sqrt{3}\right)}{33}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5\sqrt{17} times \frac{33}{33}.

$$\frac{33\times 5\sqrt{17}}{33}-\frac{16\left(6-\sqrt{3}\right)}{33}$$

Since \frac{33\times 5\sqrt{17}}{33} and \frac{16\left(6-\sqrt{3}\right)}{33} have the same denominator, subtract them by subtracting their numerators.

$$\frac{33\times 5\sqrt{17}-16\left(6-\sqrt{3}\right)}{33}$$

Do the multiplications in 33\times 5\sqrt{17}-16\left(6-\sqrt{3}\right).

$$\frac{165\sqrt{17}-96+16\sqrt{3}}{33}$$