Solve 8.10/5*frac{2sqrt{7}+3sqrt{5}}{sqrt{7}+sqrt{5}}=Psqrt{3}

$$\frac { 8.10 } { 5 } \cdot \frac { 2 \sqrt { 7 } + 3 \sqrt { 5 } } { \sqrt { 7 } + \sqrt { 5 } } = P \sqrt { 3 }$$

Solve for P

$$P = \frac{27 {(\sqrt{105} - \sqrt{3})}}{100} \approx 2.299022989$$

Steps for Solving Linear Equation

Expand \frac{8.1}{5} by multiplying both numerator and the denominator by 10.

$$\frac{81}{50}\times \frac{2\sqrt{7}+3\sqrt{5}}{\sqrt{7}+\sqrt{5}}=P\sqrt{3}$$

Rationalize the denominator of \frac{2\sqrt{7}+3\sqrt{5}}{\sqrt{7}+\sqrt{5}} by multiplying numerator and denominator by \sqrt{7}-\sqrt{5}.

$$\frac{81}{50}\times \frac{\left(2\sqrt{7}+3\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}=P\sqrt{3}$$

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

$$\frac{81}{50}\times \frac{\left(2\sqrt{7}+3\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{\left(\sqrt{7}\right)^{2}-\left(\sqrt{5}\right)^{2}}=P\sqrt{3}$$

Square \sqrt{7}. Square \sqrt{5}.

$$\frac{81}{50}\times \frac{\left(2\sqrt{7}+3\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{7-5}=P\sqrt{3}$$

Subtract 5 from 7 to get 2.

$$\frac{81}{50}\times \frac{\left(2\sqrt{7}+3\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{2}=P\sqrt{3}$$

Multiply \frac{81}{50} times \frac{\left(2\sqrt{7}+3\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{81\left(2\sqrt{7}+3\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{50\times 2}=P\sqrt{3}$$

Multiply 50 and 2 to get 100.

$$\frac{81\left(2\sqrt{7}+3\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{100}=P\sqrt{3}$$

Swap sides so that all variable terms are on the left hand side.

$$P\sqrt{3}=\frac{81\left(2\sqrt{7}+3\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{100}$$

Use the distributive property to multiply 81 by 2\sqrt{7}+3\sqrt{5}.

$$P\sqrt{3}=\frac{\left(162\sqrt{7}+243\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{100}$$

Apply the distributive property by multiplying each term of 162\sqrt{7}+243\sqrt{5} by each term of \sqrt{7}-\sqrt{5}.

$$P\sqrt{3}=\frac{162\left(\sqrt{7}\right)^{2}-162\sqrt{7}\sqrt{5}+243\sqrt{5}\sqrt{7}-243\left(\sqrt{5}\right)^{2}}{100}$$

The square of \sqrt{7} is 7.

$$P\sqrt{3}=\frac{162\times 7-162\sqrt{7}\sqrt{5}+243\sqrt{5}\sqrt{7}-243\left(\sqrt{5}\right)^{2}}{100}$$

Multiply 162 and 7 to get 1134.

$$P\sqrt{3}=\frac{1134-162\sqrt{7}\sqrt{5}+243\sqrt{5}\sqrt{7}-243\left(\sqrt{5}\right)^{2}}{100}$$

To multiply \sqrt{7} and \sqrt{5}, multiply the numbers under the square root.

$$P\sqrt{3}=\frac{1134-162\sqrt{35}+243\sqrt{5}\sqrt{7}-243\left(\sqrt{5}\right)^{2}}{100}$$

To multiply \sqrt{5} and \sqrt{7}, multiply the numbers under the square root.

$$P\sqrt{3}=\frac{1134-162\sqrt{35}+243\sqrt{35}-243\left(\sqrt{5}\right)^{2}}{100}$$

Combine -162\sqrt{35} and 243\sqrt{35} to get 81\sqrt{35}.

$$P\sqrt{3}=\frac{1134+81\sqrt{35}-243\left(\sqrt{5}\right)^{2}}{100}$$

The square of \sqrt{5} is 5.

$$P\sqrt{3}=\frac{1134+81\sqrt{35}-243\times 5}{100}$$

Multiply -243 and 5 to get -1215.

$$P\sqrt{3}=\frac{1134+81\sqrt{35}-1215}{100}$$

Subtract 1215 from 1134 to get -81.

$$P\sqrt{3}=\frac{-81+81\sqrt{35}}{100}$$

Divide each term of -81+81\sqrt{35} by 100 to get -\frac{81}{100}+\frac{81}{100}\sqrt{35}.

$$P\sqrt{3}=-\frac{81}{100}+\frac{81}{100}\sqrt{35}$$

The equation is in standard form.

$$\sqrt{3}P=\frac{81\sqrt{35}-81}{100}$$

Divide both sides by \sqrt{3}.

$$\frac{\sqrt{3}P}{\sqrt{3}}=\frac{81\sqrt{35}-81}{100\sqrt{3}}$$

Dividing by \sqrt{3} undoes the multiplication by \sqrt{3}.

$$P=\frac{81\sqrt{35}-81}{100\sqrt{3}}$$

Divide \frac{-81+81\sqrt{35}}{100} by \sqrt{3}.

$$P=\frac{27\sqrt{3}\left(\sqrt{35}-1\right)}{100}$$