Solve B=frac{sqrt{2}-sqrt{7}}{5-sqrt{8}}

$$B = \frac { \sqrt { 2 } - \sqrt { 7 } } { 5 - \sqrt { 8 } }$$

Solve for B

$$B=\frac{5\sqrt{2}+4-2\sqrt{14}-5\sqrt{7}}{17}\approx -0.567117854$$

Steps for Solving Linear Equation

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$B=\frac{\sqrt{2}-\sqrt{7}}{5-2\sqrt{2}}$$

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

$$B=\frac{\left(\sqrt{2}-\sqrt{7}\right)\left(5+2\sqrt{2}\right)}{\left(5-2\sqrt{2}\right)\left(5+2\sqrt{2}\right)}$$

Consider \left(5-2\sqrt{2}\right)\left(5+2\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}.

$$B=\frac{\left(\sqrt{2}-\sqrt{7}\right)\left(5+2\sqrt{2}\right)}{5^{2}-\left(-2\sqrt{2}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$B=\frac{\left(\sqrt{2}-\sqrt{7}\right)\left(5+2\sqrt{2}\right)}{25-\left(-2\sqrt{2}\right)^{2}}$$

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

$$B=\frac{\left(\sqrt{2}-\sqrt{7}\right)\left(5+2\sqrt{2}\right)}{25-\left(-2\right)^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate -2 to the power of 2 and get 4.

$$B=\frac{\left(\sqrt{2}-\sqrt{7}\right)\left(5+2\sqrt{2}\right)}{25-4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$B=\frac{\left(\sqrt{2}-\sqrt{7}\right)\left(5+2\sqrt{2}\right)}{25-4\times 2}$$

Multiply 4 and 2 to get 8.

$$B=\frac{\left(\sqrt{2}-\sqrt{7}\right)\left(5+2\sqrt{2}\right)}{25-8}$$

Subtract 8 from 25 to get 17.

$$B=\frac{\left(\sqrt{2}-\sqrt{7}\right)\left(5+2\sqrt{2}\right)}{17}$$

Apply the distributive property by multiplying each term of \sqrt{2}-\sqrt{7} by each term of 5+2\sqrt{2}.

$$B=\frac{5\sqrt{2}+2\left(\sqrt{2}\right)^{2}-5\sqrt{7}-2\sqrt{7}\sqrt{2}}{17}$$

The square of \sqrt{2} is 2.

$$B=\frac{5\sqrt{2}+2\times 2-5\sqrt{7}-2\sqrt{7}\sqrt{2}}{17}$$

Multiply 2 and 2 to get 4.

$$B=\frac{5\sqrt{2}+4-5\sqrt{7}-2\sqrt{7}\sqrt{2}}{17}$$

To multiply \sqrt{7} and \sqrt{2}, multiply the numbers under the square root.

$$B=\frac{5\sqrt{2}+4-5\sqrt{7}-2\sqrt{14}}{17}$$

Divide each term of 5\sqrt{2}+4-5\sqrt{7}-2\sqrt{14} by 17 to get \frac{5}{17}\sqrt{2}+\frac{4}{17}-\frac{5}{17}\sqrt{7}-\frac{2}{17}\sqrt{14}.

$$B=\frac{5}{17}\sqrt{2}+\frac{4}{17}-\frac{5}{17}\sqrt{7}-\frac{2}{17}\sqrt{14}$$

Assign B

$$B≔\frac{5\sqrt{2}+4-2\sqrt{14}-5\sqrt{7}}{17}$$