Solve sqrt{2}+1-frac{1+sqrt{2}}{sqrt{2}+156}

$$\sqrt{ 2 } +1- \frac{ 1+ \sqrt{ 2 } }{ \sqrt{ 2 } +156 }$$

Evaluate

$$\frac{24179\sqrt{2}}{24334}+\frac{12090}{12167}\approx 2.398876869$$

Short Solution Steps

Rationalize the denominator of \frac{1+\sqrt{2}}{\sqrt{2}+156} by multiplying numerator and denominator by \sqrt{2}-156.

$$\sqrt{2}+1-\frac{\left(1+\sqrt{2}\right)\left(\sqrt{2}-156\right)}{\left(\sqrt{2}+156\right)\left(\sqrt{2}-156\right)}$$

Consider \left(\sqrt{2}+156\right)\left(\sqrt{2}-156\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\sqrt{2}+1-\frac{\left(1+\sqrt{2}\right)\left(\sqrt{2}-156\right)}{\left(\sqrt{2}\right)^{2}-156^{2}}$$

Square \sqrt{2}. Square 156.

$$\sqrt{2}+1-\frac{\left(1+\sqrt{2}\right)\left(\sqrt{2}-156\right)}{2-24336}$$

Subtract 24336 from 2 to get -24334.

$$\sqrt{2}+1-\frac{\left(1+\sqrt{2}\right)\left(\sqrt{2}-156\right)}{-24334}$$

Apply the distributive property by multiplying each term of 1+\sqrt{2} by each term of \sqrt{2}-156.

$$\sqrt{2}+1-\frac{\sqrt{2}-156+\left(\sqrt{2}\right)^{2}-156\sqrt{2}}{-24334}$$

The square of \sqrt{2} is 2.

$$\sqrt{2}+1-\frac{\sqrt{2}-156+2-156\sqrt{2}}{-24334}$$

Add -156 and 2 to get -154.

$$\sqrt{2}+1-\frac{\sqrt{2}-154-156\sqrt{2}}{-24334}$$

Combine \sqrt{2} and -156\sqrt{2} to get -155\sqrt{2}.

$$\sqrt{2}+1-\frac{-155\sqrt{2}-154}{-24334}$$

Multiply both numerator and denominator by -1.

$$\sqrt{2}+1-\frac{155\sqrt{2}+154}{24334}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{2}+1 times \frac{24334}{24334}.

$$\frac{24334\left(\sqrt{2}+1\right)}{24334}-\frac{155\sqrt{2}+154}{24334}$$

Since \frac{24334\left(\sqrt{2}+1\right)}{24334} and \frac{155\sqrt{2}+154}{24334} have the same denominator, subtract them by subtracting their numerators.

$$\frac{24334\left(\sqrt{2}+1\right)-\left(155\sqrt{2}+154\right)}{24334}$$

Do the multiplications in 24334\left(\sqrt{2}+1\right)-\left(155\sqrt{2}+154\right).

$$\frac{24334\sqrt{2}[phone]\sqrt{2}-154}{24334}$$

Do the calculations in 24334\sqrt{2}[phone]\sqrt{2}-154.

$$\frac{24179\sqrt{2}+24180}{24334}$$