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

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

Laske

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

Ratkaisun vaiheet

Muunna rationaaliluvuksi nimittäjä \frac{886731088897-627013566048\sqrt{2}}{886731088897+627013566048\sqrt{2}} kertomalla osoittaja ja nimittäjä 886731088897-627013566048\sqrt{2}.

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

Tarkastele lauseketta \left(886731088897+627013566048\sqrt{2}\right)\left(886731088897-627013566048\sqrt{2}\right). Kertolasku voidaan muuntaa neliöiden erotukseksi seuraavalla säännöllä: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Kerro 886731088897-627013566048\sqrt{2} ja 886731088897-627013566048\sqrt{2}, niin saadaan \left(886731088897-627013566048\sqrt{2}\right)^{2}.

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

Käytä binomilausetta \left(a-b\right)^{2}=a^{2}-2ab+b^{2} yhtälön \left(886731088897-627013566048\sqrt{2}\right)^{2} laajentamiseen.

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

Luvun \sqrt{2} neliö on 2.

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

Kerro 393146012008229658338304 ja 2, niin saadaan 786292024016459316676608.

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

Selvitä 1572584048032918633353217 laskemalla yhteen 786292024016459316676609 ja 786292024016459316676608.

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

Laske 886731088897 potenssiin 2, jolloin ratkaisuksi tulee 786292024016459316676609.

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

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

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

Laske 627013566048 potenssiin 2, jolloin ratkaisuksi tulee 393146012008229658338304.

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

Luvun \sqrt{2} neliö on 2.

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

Kerro 393146012008229658338304 ja 2, niin saadaan 786292024016459316676608.

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

Vähennä 786292024016459316676608 luvusta 786292024016459316676609 saadaksesi tuloksen 1.

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

Luvun jakaminen yhdellä antaa tulokseksi alkuperäisen luvun.

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