Solve frac{886731088897-627013566048sqrt{2}}{886731088897+627013566048sqrt{2}}

$$\frac{ 886731088897-627013566048 \sqrt{ 2 } }{ 886731088897+627013566048 \sqrt{ 2 } }$$

Evaluate

$$1572584048032918633353217-1111984844349868137938112\sqrt{2}\approx -268435456$$

Solution Steps

Rationalize the denominator of \frac{886731088897-627013566048\sqrt{2}}{886731088897+627013566048\sqrt{2}} by multiplying numerator and denominator by 886731088897-627013566048\sqrt{2}.

$$\frac{\left(886731088897-627013566048\sqrt{2}\right)\left(886731088897-627013566048\sqrt{2}\right)}{\left(886731088897+627013566048\sqrt{2}\right)\left(886731088897-627013566048\sqrt{2}\right)}$$

Consider \left(886731088897+627013566048\sqrt{2}\right)\left(886731088897-627013566048\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}.

$$\frac{\left(886731088897-627013566048\sqrt{2}\right)\left(886731088897-627013566048\sqrt{2}\right)}{886731088897^{2}-\left(627013566048\sqrt{2}\right)^{2}}$$

Multiply 886731088897-627013566048\sqrt{2} and 886731088897-627013566048\sqrt{2} to get \left(886731088897-627013566048\sqrt{2}\right)^{2}.

$$\frac{\left(886731088897-627013566048\sqrt{2}\right)^{2}}{886731088897^{2}-\left(627013566048\sqrt{2}\right)^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(886731088897-627013566048\sqrt{2}\right)^{2}.

$$\frac{786292024016459316676609-1111984844349868137938112\sqrt{2}+393146012008229658338304\left(\sqrt{2}\right)^{2}}{886731088897^{2}-\left(627013566048\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{786292024016459316676609-1111984844349868137938112\sqrt{2}+393146012008229658338304\times 2}{886731088897^{2}-\left(627013566048\sqrt{2}\right)^{2}}$$

Multiply 393146012008229658338304 and 2 to get 786292024016459316676608.

$$\frac{786292024016459316676609-1111984844349868137938112\sqrt{2}+786292024016459316676608}{886731088897^{2}-\left(627013566048\sqrt{2}\right)^{2}}$$

Add 786292024016459316676609 and 786292024016459316676608 to get 1572584048032918633353217.

$$\frac{1572584048032918633353217-1111984844349868137938112\sqrt{2}}{886731088897^{2}-\left(627013566048\sqrt{2}\right)^{2}}$$

Calculate 886731088897 to the power of 2 and get 786292024016459316676609.

$$\frac{1572584048032918633353217-1111984844349868137938112\sqrt{2}}{786292024016459316676609-\left(627013566048\sqrt{2}\right)^{2}}$$

Expand \left(627013566048\sqrt{2}\right)^{2}.

$$\frac{1572584048032918633353217-1111984844349868137938112\sqrt{2}}{786292024016459316676609-627013566048^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 627013566048 to the power of 2 and get 393146012008229658338304.

$$\frac{1572584048032918633353217-1111984844349868137938112\sqrt{2}}{786292024016459316676609-393146012008229658338304\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{1572584048032918633353217-1111984844349868137938112\sqrt{2}}{786292024016459316676609-393146012008229658338304\times 2}$$

Multiply 393146012008229658338304 and 2 to get 786292024016459316676608.

$$\frac{1572584048032918633353217-1111984844349868137938112\sqrt{2}}{786292024016459316676609-786292024016459316676608}$$

Subtract 786292024016459316676608 from 786292024016459316676609 to get 1.

$$\frac{1572584048032918633353217-1111984844349868137938112\sqrt{2}}{1}$$

Anything divided by one gives itself.

$$1572584048032918633353217-1111984844349868137938112\sqrt{2}$$