Solve frac{sqrt{2}+1}{(sqrt{2}-1)(sqrt{2}+5)}-frac{sqrt{2}-1}{2sqrt{2}(sqrt{2}+1)}

$$\frac { \sqrt { 2 } + 1 } { ( \sqrt { 2 } - 1 ) ( \sqrt { 2 } + 5 ) } - \frac { \sqrt { 2 } - 1 } { 2 \sqrt { 2 } ( \sqrt { 2 } + 1 ) }$$

Evaluate

$$-\frac{41\sqrt{2}}{92}+\frac{34}{23}\approx 0.848013521$$

Short Solution Steps

Rationalize the denominator of \frac{\sqrt{2}-1}{2\sqrt{2}\left(\sqrt{2}+1\right)} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{2}+1}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+5\right)}-\frac{\left(\sqrt{2}-1\right)\sqrt{2}}{2\left(\sqrt{2}\right)^{2}\left(\sqrt{2}+1\right)}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}+1}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+5\right)}-\frac{\left(\sqrt{2}-1\right)\sqrt{2}}{2\times 2\left(\sqrt{2}+1\right)}$$

Multiply 2 and 2 to get 4.

$$\frac{\sqrt{2}+1}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+5\right)}-\frac{\left(\sqrt{2}-1\right)\sqrt{2}}{4\left(\sqrt{2}+1\right)}$$

Apply the distributive property by multiplying each term of \sqrt{2}-1 by each term of \sqrt{2}+5.

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

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}+1}{2+5\sqrt{2}-\sqrt{2}-5}-\frac{\left(\sqrt{2}-1\right)\sqrt{2}}{4\left(\sqrt{2}+1\right)}$$

Combine 5\sqrt{2} and -\sqrt{2} to get 4\sqrt{2}.

$$\frac{\sqrt{2}+1}{2+4\sqrt{2}-5}-\frac{\left(\sqrt{2}-1\right)\sqrt{2}}{4\left(\sqrt{2}+1\right)}$$

Subtract 5 from 2 to get -3.

$$\frac{\sqrt{2}+1}{-3+4\sqrt{2}}-\frac{\left(\sqrt{2}-1\right)\sqrt{2}}{4\left(\sqrt{2}+1\right)}$$

Rationalize the denominator of \frac{\sqrt{2}+1}{-3+4\sqrt{2}} by multiplying numerator and denominator by -3-4\sqrt{2}.

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{\left(-3+4\sqrt{2}\right)\left(-3-4\sqrt{2}\right)}-\frac{\left(\sqrt{2}-1\right)\sqrt{2}}{4\left(\sqrt{2}+1\right)}$$

Consider \left(-3+4\sqrt{2}\right)\left(-3-4\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Calculate -3 to the power of 2 and get 9.

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{9-\left(4\sqrt{2}\right)^{2}}-\frac{\left(\sqrt{2}-1\right)\sqrt{2}}{4\left(\sqrt{2}+1\right)}$$

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

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{9-4^{2}\left(\sqrt{2}\right)^{2}}-\frac{\left(\sqrt{2}-1\right)\sqrt{2}}{4\left(\sqrt{2}+1\right)}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{9-16\left(\sqrt{2}\right)^{2}}-\frac{\left(\sqrt{2}-1\right)\sqrt{2}}{4\left(\sqrt{2}+1\right)}$$

The square of \sqrt{2} is 2.

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{9-16\times 2}-\frac{\left(\sqrt{2}-1\right)\sqrt{2}}{4\left(\sqrt{2}+1\right)}$$

Multiply 16 and 2 to get 32.

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{9-32}-\frac{\left(\sqrt{2}-1\right)\sqrt{2}}{4\left(\sqrt{2}+1\right)}$$

Subtract 32 from 9 to get -23.

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{-23}-\frac{\left(\sqrt{2}-1\right)\sqrt{2}}{4\left(\sqrt{2}+1\right)}$$

Use the distributive property to multiply \sqrt{2}-1 by \sqrt{2}.

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{-23}-\frac{\left(\sqrt{2}\right)^{2}-\sqrt{2}}{4\left(\sqrt{2}+1\right)}$$

The square of \sqrt{2} is 2.

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{-23}-\frac{2-\sqrt{2}}{4\left(\sqrt{2}+1\right)}$$

Use the distributive property to multiply 4 by \sqrt{2}+1.

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{-23}-\frac{2-\sqrt{2}}{4\sqrt{2}+4}$$

Rationalize the denominator of \frac{2-\sqrt{2}}{4\sqrt{2}+4} by multiplying numerator and denominator by 4\sqrt{2}-4.

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{-23}-\frac{\left(2-\sqrt{2}\right)\left(4\sqrt{2}-4\right)}{\left(4\sqrt{2}+4\right)\left(4\sqrt{2}-4\right)}$$

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

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{-23}-\frac{\left(2-\sqrt{2}\right)\left(4\sqrt{2}-4\right)}{\left(4\sqrt{2}\right)^{2}-4^{2}}$$

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

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{-23}-\frac{\left(2-\sqrt{2}\right)\left(4\sqrt{2}-4\right)}{4^{2}\left(\sqrt{2}\right)^{2}-4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{-23}-\frac{\left(2-\sqrt{2}\right)\left(4\sqrt{2}-4\right)}{16\left(\sqrt{2}\right)^{2}-4^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{-23}-\frac{\left(2-\sqrt{2}\right)\left(4\sqrt{2}-4\right)}{16\times 2-4^{2}}$$

Multiply 16 and 2 to get 32.

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{-23}-\frac{\left(2-\sqrt{2}\right)\left(4\sqrt{2}-4\right)}{32-4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{-23}-\frac{\left(2-\sqrt{2}\right)\left(4\sqrt{2}-4\right)}{32-16}$$

Subtract 16 from 32 to get 16.

$$\frac{\left(\sqrt{2}+1\right)\left(-3-4\sqrt{2}\right)}{-23}-\frac{\left(2-\sqrt{2}\right)\left(4\sqrt{2}-4\right)}{16}$$

Apply the distributive property by multiplying each term of \sqrt{2}+1 by each term of -3-4\sqrt{2}.

$$\frac{-3\sqrt{2}-4\left(\sqrt{2}\right)^{2}-3-4\sqrt{2}}{-23}-\frac{\left(2-\sqrt{2}\right)\left(4\sqrt{2}-4\right)}{16}$$

The square of \sqrt{2} is 2.

$$\frac{-3\sqrt{2}-4\times 2-3-4\sqrt{2}}{-23}-\frac{\left(2-\sqrt{2}\right)\left(4\sqrt{2}-4\right)}{16}$$

Multiply -4 and 2 to get -8.

$$\frac{-3\sqrt{2}-8-3-4\sqrt{2}}{-23}-\frac{\left(2-\sqrt{2}\right)\left(4\sqrt{2}-4\right)}{16}$$

Subtract 3 from -8 to get -11.

$$\frac{-3\sqrt{2}-11-4\sqrt{2}}{-23}-\frac{\left(2-\sqrt{2}\right)\left(4\sqrt{2}-4\right)}{16}$$

Combine -3\sqrt{2} and -4\sqrt{2} to get -7\sqrt{2}.

$$\frac{-7\sqrt{2}-11}{-23}-\frac{\left(2-\sqrt{2}\right)\left(4\sqrt{2}-4\right)}{16}$$

Multiply both numerator and denominator by -1.

$$\frac{7\sqrt{2}+11}{23}-\frac{\left(2-\sqrt{2}\right)\left(4\sqrt{2}-4\right)}{16}$$

Apply the distributive property by multiplying each term of 2-\sqrt{2} by each term of 4\sqrt{2}-4.

$$\frac{7\sqrt{2}+11}{23}-\frac{8\sqrt{2}-8-4\left(\sqrt{2}\right)^{2}+4\sqrt{2}}{16}$$

The square of \sqrt{2} is 2.

$$\frac{7\sqrt{2}+11}{23}-\frac{8\sqrt{2}-8-4\times 2+4\sqrt{2}}{16}$$

Multiply -4 and 2 to get -8.

$$\frac{7\sqrt{2}+11}{23}-\frac{8\sqrt{2}-8-8+4\sqrt{2}}{16}$$

Subtract 8 from -8 to get -16.

$$\frac{7\sqrt{2}+11}{23}-\frac{8\sqrt{2}-16+4\sqrt{2}}{16}$$

Combine 8\sqrt{2} and 4\sqrt{2} to get 12\sqrt{2}.

$$\frac{7\sqrt{2}+11}{23}-\frac{12\sqrt{2}-16}{16}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 23 and 16 is 368. Multiply \frac{7\sqrt{2}+11}{23} times \frac{16}{16}. Multiply \frac{12\sqrt{2}-16}{16} times \frac{23}{23}.

$$\frac{16\left(7\sqrt{2}+11\right)}{368}-\frac{23\left(12\sqrt{2}-16\right)}{368}$$

Since \frac{16\left(7\sqrt{2}+11\right)}{368} and \frac{23\left(12\sqrt{2}-16\right)}{368} have the same denominator, subtract them by subtracting their numerators.

$$\frac{16\left(7\sqrt{2}+11\right)-23\left(12\sqrt{2}-16\right)}{368}$$

Do the multiplications in 16\left(7\sqrt{2}+11\right)-23\left(12\sqrt{2}-16\right).

$$\frac{112\sqrt{2}+176-276\sqrt{2}+368}{368}$$

Do the calculations in 112\sqrt{2}+176-276\sqrt{2}+368.

$$\frac{-164\sqrt{2}+544}{368}$$

Factor

$$\frac{136 - 41 \sqrt{2}}{92} = 0.8480135211163379$$