Lahenda frac{sqrt{-8}+1}{sqrt{-8}-1}

$$\frac { \sqrt { - 8 } + 1 } { \sqrt { - 8 } - 1 }$$

Arvuta (complex solution)

$$\frac{-4\sqrt{2}i+7}{9}\approx 0,777777778-0,628539361i$$

Lahenduse etapid

Tegurda -8=\left(2i\right)^{2}\times 2. Kirjutage \sqrt{\left(2i\right)^{2}\times 2} toote juured, kui see ruut \sqrt{\left(2i\right)^{2}}\sqrt{2}. Leidke \left(2i\right)^{2} ruutjuur.

$$\frac{2i\sqrt{2}+1}{\sqrt{-8}-1}$$

Tegurda -8=\left(2i\right)^{2}\times 2. Kirjutage \sqrt{\left(2i\right)^{2}\times 2} toote juured, kui see ruut \sqrt{\left(2i\right)^{2}}\sqrt{2}. Leidke \left(2i\right)^{2} ruutjuur.

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

Ratsionaliseerige korrutades lugeja ja 2i\sqrt{2}+1 nimetaja \frac{2i\sqrt{2}+1}{2i\sqrt{2}-1} nimetaja.

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

Mõelge valemile \left(2i\sqrt{2}-1\right)\left(2i\sqrt{2}+1\right). Korrutustehte saab ruutude vaheks teisendada järgmise reegli abil: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Korrutage 2i\sqrt{2}+1 ja 2i\sqrt{2}+1, et leida \left(2i\sqrt{2}+1\right)^{2}.

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

Kasutage kaksliikme \left(2i\sqrt{2}+1\right)^{2} arendamiseks binoomvalemit \left(a+b\right)^{2}=a^{2}+2ab+b^{2}.

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

\sqrt{2} ruut on 2.

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

Korrutage -4 ja 2, et leida -8.

$$\frac{-8+4i\sqrt{2}+1}{\left(2i\sqrt{2}\right)^{2}-1^{2}}$$

Liitke -8 ja 1, et leida -7.

$$\frac{-7+4i\sqrt{2}}{\left(2i\sqrt{2}\right)^{2}-1^{2}}$$

Laiendage \left(2i\sqrt{2}\right)^{2}.

$$\frac{-7+4i\sqrt{2}}{\left(2i\right)^{2}\left(\sqrt{2}\right)^{2}-1^{2}}$$

Arvutage 2 aste 2i ja leidke -4.

$$\frac{-7+4i\sqrt{2}}{-4\left(\sqrt{2}\right)^{2}-1^{2}}$$

\sqrt{2} ruut on 2.

$$\frac{-7+4i\sqrt{2}}{-4\times 2-1^{2}}$$

Korrutage -4 ja 2, et leida -8.

$$\frac{-7+4i\sqrt{2}}{-8-1^{2}}$$

Arvutage 2 aste 1 ja leidke 1.

$$\frac{-7+4i\sqrt{2}}{-8-1}$$

Lahutage 1 väärtusest -8, et leida -9.

$$\frac{-7+4i\sqrt{2}}{-9}$$

Korrutage nii lugeja kui ka nimetaja arvuga –1.

$$\frac{7-4i\sqrt{2}}{9}$$

Reaalosa (complex solution)

$$\frac{7}{9} = 0,7777777777777778$$

Arvuta

$$\text{Indeterminate}$$