Solve frac{frac{155sqrt{3}}{4}}{frac{26+sqrt{181}}{2}}

$$\frac { \frac { 155 \sqrt { 3 } } { 4 } } { \frac { 26 + \sqrt { 181 } } { 2 } }$$

Evaluate

$$-\frac{31\sqrt{3}\left(\sqrt{181}-26\right)}{198}\approx 3.402322114$$

Solution Steps

Divide \frac{155\sqrt{3}}{4} by \frac{26+\sqrt{181}}{2} by multiplying \frac{155\sqrt{3}}{4} by the reciprocal of \frac{26+\sqrt{181}}{2}.

$$\frac{155\sqrt{3}\times 2}{4\left(26+\sqrt{181}\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{155\sqrt{3}}{2\left(\sqrt{181}+26\right)}$$

Use the distributive property to multiply 2 by \sqrt{181}+26.

$$\frac{155\sqrt{3}}{2\sqrt{181}+52}$$

Rationalize the denominator of \frac{155\sqrt{3}}{2\sqrt{181}+52} by multiplying numerator and denominator by 2\sqrt{181}-52.

$$\frac{155\sqrt{3}\left(2\sqrt{181}-52\right)}{\left(2\sqrt{181}+52\right)\left(2\sqrt{181}-52\right)}$$

Consider \left(2\sqrt{181}+52\right)\left(2\sqrt{181}-52\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{155\sqrt{3}\left(2\sqrt{181}-52\right)}{\left(2\sqrt{181}\right)^{2}-52^{2}}$$

Expand \left(2\sqrt{181}\right)^{2}.

$$\frac{155\sqrt{3}\left(2\sqrt{181}-52\right)}{2^{2}\left(\sqrt{181}\right)^{2}-52^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{155\sqrt{3}\left(2\sqrt{181}-52\right)}{4\left(\sqrt{181}\right)^{2}-52^{2}}$$

The square of \sqrt{181} is 181.

$$\frac{155\sqrt{3}\left(2\sqrt{181}-52\right)}{4\times 181-52^{2}}$$

Multiply 4 and 181 to get 724.

$$\frac{155\sqrt{3}\left(2\sqrt{181}-52\right)}{724-52^{2}}$$

Calculate 52 to the power of 2 and get 2704.

$$\frac{155\sqrt{3}\left(2\sqrt{181}-52\right)}{724-2704}$$

Subtract 2704 from 724 to get -1980.

$$\frac{155\sqrt{3}\left(2\sqrt{181}-52\right)}{-1980}$$

Divide 155\sqrt{3}\left(2\sqrt{181}-52\right) by -1980 to get -\frac{31}{396}\sqrt{3}\left(2\sqrt{181}-52\right).

$$-\frac{31}{396}\sqrt{3}\left(2\sqrt{181}-52\right)$$

Use the distributive property to multiply -\frac{31}{396}\sqrt{3} by 2\sqrt{181}-52.

$$-\frac{31}{396}\sqrt{3}\times 2\sqrt{181}-\frac{31}{396}\sqrt{3}\left(-52\right)$$

Express -\frac{31}{396}\times 2 as a single fraction.

$$\frac{-31\times 2}{396}\sqrt{3}\sqrt{181}-\frac{31}{396}\sqrt{3}\left(-52\right)$$

Multiply -31 and 2 to get -62.

$$\frac{-62}{396}\sqrt{3}\sqrt{181}-\frac{31}{396}\sqrt{3}\left(-52\right)$$

Reduce the fraction \frac{-62}{396} to lowest terms by extracting and canceling out 2.

$$-\frac{31}{198}\sqrt{3}\sqrt{181}-\frac{31}{396}\sqrt{3}\left(-52\right)$$

To multiply \sqrt{3} and \sqrt{181}, multiply the numbers under the square root.

$$-\frac{31}{198}\sqrt{543}-\frac{31}{396}\sqrt{3}\left(-52\right)$$

Express -\frac{31}{396}\left(-52\right) as a single fraction.

$$-\frac{31}{198}\sqrt{543}+\frac{-31\left(-52\right)}{396}\sqrt{3}$$

Multiply -31 and -52 to get 1612.

$$-\frac{31}{198}\sqrt{543}+\frac{1612}{396}\sqrt{3}$$

Reduce the fraction \frac{1612}{396} to lowest terms by extracting and canceling out 4.

$$-\frac{31}{198}\sqrt{543}+\frac{403}{99}\sqrt{3}$$