Solve frac{sqrt{7}+3sqrt{11}}{5sqrt{7}+4sqrt{11}}frac{5sqrt{7}-4sqrt{11}}{5sqrt{7}-4sqrt{11}}

$$\frac{ \sqrt{ 7 } +3 \sqrt{ 11 } }{ 5 \sqrt{ 7 } +4 \sqrt{ 11 } } \frac{ 5 \sqrt{ 7 } -4 \sqrt{ 11 } }{ 5 \sqrt{ 7 } -4 \sqrt{ 11 } }$$

Evaluate

$$97-11\sqrt{77}\approx 0.475391739$$

Solution Steps

Divide 5\sqrt{7}-4\sqrt{11} by 5\sqrt{7}-4\sqrt{11} to get 1.

$$\frac{\sqrt{7}+3\sqrt{11}}{5\sqrt{7}+4\sqrt{11}}\times 1$$

Rationalize the denominator of \frac{\sqrt{7}+3\sqrt{11}}{5\sqrt{7}+4\sqrt{11}} by multiplying numerator and denominator by 5\sqrt{7}-4\sqrt{11}.

$$\frac{\left(\sqrt{7}+3\sqrt{11}\right)\left(5\sqrt{7}-4\sqrt{11}\right)}{\left(5\sqrt{7}+4\sqrt{11}\right)\left(5\sqrt{7}-4\sqrt{11}\right)}\times 1$$

Consider \left(5\sqrt{7}+4\sqrt{11}\right)\left(5\sqrt{7}-4\sqrt{11}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{7}+3\sqrt{11}\right)\left(5\sqrt{7}-4\sqrt{11}\right)}{\left(5\sqrt{7}\right)^{2}-\left(4\sqrt{11}\right)^{2}}\times 1$$

Expand \left(5\sqrt{7}\right)^{2}.

$$\frac{\left(\sqrt{7}+3\sqrt{11}\right)\left(5\sqrt{7}-4\sqrt{11}\right)}{5^{2}\left(\sqrt{7}\right)^{2}-\left(4\sqrt{11}\right)^{2}}\times 1$$

Calculate 5 to the power of 2 and get 25.

$$\frac{\left(\sqrt{7}+3\sqrt{11}\right)\left(5\sqrt{7}-4\sqrt{11}\right)}{25\left(\sqrt{7}\right)^{2}-\left(4\sqrt{11}\right)^{2}}\times 1$$

The square of \sqrt{7} is 7.

$$\frac{\left(\sqrt{7}+3\sqrt{11}\right)\left(5\sqrt{7}-4\sqrt{11}\right)}{25\times 7-\left(4\sqrt{11}\right)^{2}}\times 1$$

Multiply 25 and 7 to get 175.

$$\frac{\left(\sqrt{7}+3\sqrt{11}\right)\left(5\sqrt{7}-4\sqrt{11}\right)}{175-\left(4\sqrt{11}\right)^{2}}\times 1$$

Expand \left(4\sqrt{11}\right)^{2}.

$$\frac{\left(\sqrt{7}+3\sqrt{11}\right)\left(5\sqrt{7}-4\sqrt{11}\right)}{175-4^{2}\left(\sqrt{11}\right)^{2}}\times 1$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(\sqrt{7}+3\sqrt{11}\right)\left(5\sqrt{7}-4\sqrt{11}\right)}{175-16\left(\sqrt{11}\right)^{2}}\times 1$$

The square of \sqrt{11} is 11.

$$\frac{\left(\sqrt{7}+3\sqrt{11}\right)\left(5\sqrt{7}-4\sqrt{11}\right)}{175-16\times 11}\times 1$$

Multiply 16 and 11 to get 176.

$$\frac{\left(\sqrt{7}+3\sqrt{11}\right)\left(5\sqrt{7}-4\sqrt{11}\right)}{175-176}\times 1$$

Subtract 176 from 175 to get -1.

$$\frac{\left(\sqrt{7}+3\sqrt{11}\right)\left(5\sqrt{7}-4\sqrt{11}\right)}{-1}\times 1$$

Anything divided by -1 gives its opposite.

$$\left(-\left(\sqrt{7}+3\sqrt{11}\right)\left(5\sqrt{7}-4\sqrt{11}\right)\right)\times 1$$

Apply the distributive property by multiplying each term of \sqrt{7}+3\sqrt{11} by each term of 5\sqrt{7}-4\sqrt{11}.

$$\left(-\left(5\left(\sqrt{7}\right)^{2}-4\sqrt{7}\sqrt{11}+15\sqrt{11}\sqrt{7}-12\left(\sqrt{11}\right)^{2}\right)\right)\times 1$$

The square of \sqrt{7} is 7.

$$\left(-\left(5\times 7-4\sqrt{7}\sqrt{11}+15\sqrt{11}\sqrt{7}-12\left(\sqrt{11}\right)^{2}\right)\right)\times 1$$

Multiply 5 and 7 to get 35.

$$\left(-\left(35-4\sqrt{7}\sqrt{11}+15\sqrt{11}\sqrt{7}-12\left(\sqrt{11}\right)^{2}\right)\right)\times 1$$

To multiply \sqrt{7} and \sqrt{11}, multiply the numbers under the square root.

$$\left(-\left(35-4\sqrt{77}+15\sqrt{11}\sqrt{7}-12\left(\sqrt{11}\right)^{2}\right)\right)\times 1$$

To multiply \sqrt{11} and \sqrt{7}, multiply the numbers under the square root.

$$\left(-\left(35-4\sqrt{77}+15\sqrt{77}-12\left(\sqrt{11}\right)^{2}\right)\right)\times 1$$

Combine -4\sqrt{77} and 15\sqrt{77} to get 11\sqrt{77}.

$$\left(-\left(35+11\sqrt{77}-12\left(\sqrt{11}\right)^{2}\right)\right)\times 1$$

The square of \sqrt{11} is 11.

$$\left(-\left(35+11\sqrt{77}-12\times 11\right)\right)\times 1$$

Multiply -12 and 11 to get -132.

$$\left(-\left(35+11\sqrt{77}-132\right)\right)\times 1$$

Subtract 132 from 35 to get -97.

$$\left(-\left(-97+11\sqrt{77}\right)\right)\times 1$$

To find the opposite of -97+11\sqrt{77}, find the opposite of each term.

$$\left(-\left(-97\right)-11\sqrt{77}\right)\times 1$$

The opposite of -97 is 97.

$$\left(97-11\sqrt{77}\right)\times 1$$

Use the distributive property to multiply 97-11\sqrt{77} by 1.

$$97-11\sqrt{77}$$