Réitigh (1+i/1-i)^6+frac{sqrt{2}+sqrt{3}i}{sqrt{3}-sqrt{2}i}

$$( \frac { 1 + i } { 1 - i } ) ^ { 6 } + \frac { \sqrt { 2 } + \sqrt { 3 } i } { \sqrt { 3 } - \sqrt { 2 } i }$$

Luacháil

$$-1+i$$

Céimeanna Réitigh

Iolraigh uimhreoir agus ainmneoir \frac{1+i}{1-i} faoi chomhchuingeach coimpléascach an ainmneora, 1+i.

$$\left(\frac{\left(1+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}\right)^{6}+\frac{\sqrt{2}+i\sqrt{3}}{\sqrt{3}-i\sqrt{2}}$$

Déan iolrúcháin in \frac{\left(1+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}.

$$\left(\frac{2i}{2}\right)^{6}+\frac{\sqrt{2}+i\sqrt{3}}{\sqrt{3}-i\sqrt{2}}$$

Roinn 2i faoi 2 chun i a fháil.

$$i^{6}+\frac{\sqrt{2}+i\sqrt{3}}{\sqrt{3}-i\sqrt{2}}$$

Ríomh cumhacht i de 6 agus faigh -1.

$$-1+\frac{\sqrt{2}+i\sqrt{3}}{\sqrt{3}-i\sqrt{2}}$$

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi \sqrt{3}+i\sqrt{2} chun ainmneoir \frac{\sqrt{2}+i\sqrt{3}}{\sqrt{3}-i\sqrt{2}} a thiontú in uimhir chóimheasta.

$$-1+\frac{\left(\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{3}+i\sqrt{2}\right)}{\left(\sqrt{3}-i\sqrt{2}\right)\left(\sqrt{3}+i\sqrt{2}\right)}$$

Mar shampla \left(\sqrt{3}-i\sqrt{2}\right)\left(\sqrt{3}+i\sqrt{2}\right). Is féidir iolrúchán a athrú ó bhonn go dtí difríocht na gcearnóg ag úsáid na rialach seo: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$-1+\frac{\left(\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{3}+i\sqrt{2}\right)}{\left(\sqrt{3}\right)^{2}-\left(-i\sqrt{2}\right)^{2}}$$

Is é 3 uimhir chearnach \sqrt{3}.

$$-1+\frac{\left(\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{3}+i\sqrt{2}\right)}{3-\left(-i\sqrt{2}\right)^{2}}$$

Fairsingigh \left(-i\sqrt{2}\right)^{2}

$$-1+\frac{\left(\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{3}+i\sqrt{2}\right)}{3-\left(-i\right)^{2}\left(\sqrt{2}\right)^{2}}$$

Ríomh cumhacht -i de 2 agus faigh -1.

$$-1+\frac{\left(\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{3}+i\sqrt{2}\right)}{3-\left(-\left(\sqrt{2}\right)^{2}\right)}$$

Is é 2 uimhir chearnach \sqrt{2}.

$$-1+\frac{\left(\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{3}+i\sqrt{2}\right)}{3-\left(-2\right)}$$

Méadaigh -1 agus -2 chun 2 a fháil.

$$-1+\frac{\left(\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{3}+i\sqrt{2}\right)}{3+2}$$

Suimigh 3 agus 2 chun 5 a fháil.

$$-1+\frac{\left(\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{3}+i\sqrt{2}\right)}{5}$$

Úsáid an t-airí dáileach chun \sqrt{2}+i\sqrt{3} a mhéadú faoi \sqrt{3}+i\sqrt{2} agus chun téarmaí comhchosúla a chumasc.

$$-1+\frac{i\left(\sqrt{2}\right)^{2}+i\left(\sqrt{3}\right)^{2}}{5}$$

Is é 2 uimhir chearnach \sqrt{2}.

$$-1+\frac{2i+i\left(\sqrt{3}\right)^{2}}{5}$$

Is é 3 uimhir chearnach \sqrt{3}.

$$-1+\frac{2i+3i}{5}$$

Suimigh 2i agus 3i chun 5i a fháil.

$$-1+\frac{5i}{5}$$

Roinn 5i faoi 5 chun i a fháil.

$$-1+i$$

Fíorpháirt

$$-1$$

Céimeanna Réitigh

Iolraigh uimhreoir agus ainmneoir \frac{1+i}{1-i} faoi chomhchuingeach coimpléascach an ainmneora, 1+i.

$$Re(\left(\frac{\left(1+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}\right)^{6}+\frac{\sqrt{2}+i\sqrt{3}}{\sqrt{3}-i\sqrt{2}})$$

Déan iolrúcháin in \frac{\left(1+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}.

$$Re(\left(\frac{2i}{2}\right)^{6}+\frac{\sqrt{2}+i\sqrt{3}}{\sqrt{3}-i\sqrt{2}})$$

Roinn 2i faoi 2 chun i a fháil.

$$Re(i^{6}+\frac{\sqrt{2}+i\sqrt{3}}{\sqrt{3}-i\sqrt{2}})$$

Ríomh cumhacht i de 6 agus faigh -1.

$$Re(-1+\frac{\sqrt{2}+i\sqrt{3}}{\sqrt{3}-i\sqrt{2}})$$

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi \sqrt{3}+i\sqrt{2} chun ainmneoir \frac{\sqrt{2}+i\sqrt{3}}{\sqrt{3}-i\sqrt{2}} a thiontú in uimhir chóimheasta.

$$Re(-1+\frac{\left(\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{3}+i\sqrt{2}\right)}{\left(\sqrt{3}-i\sqrt{2}\right)\left(\sqrt{3}+i\sqrt{2}\right)})$$

Mar shampla \left(\sqrt{3}-i\sqrt{2}\right)\left(\sqrt{3}+i\sqrt{2}\right). Is féidir iolrúchán a athrú ó bhonn go dtí difríocht na gcearnóg ag úsáid na rialach seo: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$Re(-1+\frac{\left(\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{3}+i\sqrt{2}\right)}{\left(\sqrt{3}\right)^{2}-\left(-i\sqrt{2}\right)^{2}})$$

Is é 3 uimhir chearnach \sqrt{3}.

$$Re(-1+\frac{\left(\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{3}+i\sqrt{2}\right)}{3-\left(-i\sqrt{2}\right)^{2}})$$

Fairsingigh \left(-i\sqrt{2}\right)^{2}

$$Re(-1+\frac{\left(\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{3}+i\sqrt{2}\right)}{3-\left(-i\right)^{2}\left(\sqrt{2}\right)^{2}})$$

Ríomh cumhacht -i de 2 agus faigh -1.

$$Re(-1+\frac{\left(\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{3}+i\sqrt{2}\right)}{3-\left(-\left(\sqrt{2}\right)^{2}\right)})$$

Is é 2 uimhir chearnach \sqrt{2}.

$$Re(-1+\frac{\left(\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{3}+i\sqrt{2}\right)}{3-\left(-2\right)})$$

Méadaigh -1 agus -2 chun 2 a fháil.

$$Re(-1+\frac{\left(\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{3}+i\sqrt{2}\right)}{3+2})$$

Suimigh 3 agus 2 chun 5 a fháil.

$$Re(-1+\frac{\left(\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{3}+i\sqrt{2}\right)}{5})$$

Úsáid an t-airí dáileach chun \sqrt{2}+i\sqrt{3} a mhéadú faoi \sqrt{3}+i\sqrt{2} agus chun téarmaí comhchosúla a chumasc.

$$Re(-1+\frac{i\left(\sqrt{2}\right)^{2}+i\left(\sqrt{3}\right)^{2}}{5})$$

Is é 2 uimhir chearnach \sqrt{2}.

$$Re(-1+\frac{2i+i\left(\sqrt{3}\right)^{2}}{5})$$

Is é 3 uimhir chearnach \sqrt{3}.

$$Re(-1+\frac{2i+3i}{5})$$

Suimigh 2i agus 3i chun 5i a fháil.

$$Re(-1+\frac{5i}{5})$$

Roinn 5i faoi 5 chun i a fháil.

$$Re(-1+i)$$

Is é -1 fíorchuid -1+i.

$$-1$$