Solve 4/55+433sqrt{frac{55{36}}}

$$\frac{ 4 }{ 55+433 \sqrt{ \frac{ 55 }{ 36 } } }$$

Evaluate

$$\frac{10392\sqrt{55}}{10202995}-\frac{144}{185509}\approx 0.006777337$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{55}{36}} as the division of square roots \frac{\sqrt{55}}{\sqrt{36}}.

$$\frac{4}{55+433\times \frac{\sqrt{55}}{\sqrt{36}}}$$

Calculate the square root of 36 and get 6.

$$\frac{4}{55+433\times \frac{\sqrt{55}}{6}}$$

Express 433\times \frac{\sqrt{55}}{6} as a single fraction.

$$\frac{4}{55+\frac{433\sqrt{55}}{6}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 55 times \frac{6}{6}.

$$\frac{4}{\frac{55\times 6}{6}+\frac{433\sqrt{55}}{6}}$$

Since \frac{55\times 6}{6} and \frac{433\sqrt{55}}{6} have the same denominator, add them by adding their numerators.

$$\frac{4}{\frac{55\times 6+433\sqrt{55}}{6}}$$

Do the multiplications in 55\times 6+433\sqrt{55}.

$$\frac{4}{\frac{330+433\sqrt{55}}{6}}$$

Divide 4 by \frac{330+433\sqrt{55}}{6} by multiplying 4 by the reciprocal of \frac{330+433\sqrt{55}}{6}.

$$\frac{4\times 6}{330+433\sqrt{55}}$$

Rationalize the denominator of \frac{4\times 6}{330+433\sqrt{55}} by multiplying numerator and denominator by 330-433\sqrt{55}.

$$\frac{4\times 6\left(330-433\sqrt{55}\right)}{\left(330+433\sqrt{55}\right)\left(330-433\sqrt{55}\right)}$$

Consider \left(330+433\sqrt{55}\right)\left(330-433\sqrt{55}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{4\times 6\left(330-433\sqrt{55}\right)}{330^{2}-\left(433\sqrt{55}\right)^{2}}$$

Multiply 4 and 6 to get 24.

$$\frac{24\left(330-433\sqrt{55}\right)}{330^{2}-\left(433\sqrt{55}\right)^{2}}$$

Calculate 330 to the power of 2 and get 108900.

$$\frac{24\left(330-433\sqrt{55}\right)}{108900-\left(433\sqrt{55}\right)^{2}}$$

Expand \left(433\sqrt{55}\right)^{2}.

$$\frac{24\left(330-433\sqrt{55}\right)}{108900-433^{2}\left(\sqrt{55}\right)^{2}}$$

Calculate 433 to the power of 2 and get 187489.

$$\frac{24\left(330-433\sqrt{55}\right)}{108900-187489\left(\sqrt{55}\right)^{2}}$$

The square of \sqrt{55} is 55.

$$\frac{24\left(330-433\sqrt{55}\right)}{108900-187489\times 55}$$

Multiply 187489 and 55 to get 10311895.

$$\frac{24\left(330-433\sqrt{55}\right)}{108900-10311895}$$

Subtract 10311895 from 108900 to get -10202995.

$$\frac{24\left(330-433\sqrt{55}\right)}{-10202995}$$

Use the distributive property to multiply 24 by 330-433\sqrt{55}.

$$\frac{7920-10392\sqrt{55}}{-10202995}$$