Réitigh frac{2+sqrt{3}}{2-sqrt{3}}+frac{2-sqrt{3}}{2+sqrt{3}}+frac{sqrt{3}+1}{sqrt{3}-1}

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

Luacháil

$$\sqrt{3}+16\approx 17.732050808$$

Céimeanna Réitigh

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

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

Mar shampla \left(2-\sqrt{3}\right)\left(2+\sqrt{3}\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}.

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

Cearnóg 2. Cearnóg \sqrt{3}.

$$\frac{\left(2+\sqrt{3}\right)\left(2+\sqrt{3}\right)}{4-3}+\frac{2-\sqrt{3}}{2+\sqrt{3}}+\frac{\sqrt{3}+1}{\sqrt{3}-1}$$

Dealaigh 3 ó 4 chun 1 a fháil.

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

Tugann aon rud a roinntear ar a haon é féin.

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

Méadaigh 2+\sqrt{3} agus 2+\sqrt{3} chun \left(2+\sqrt{3}\right)^{2} a fháil.

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

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

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

Mar shampla \left(2+\sqrt{3}\right)\left(2-\sqrt{3}\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}.

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

Cearnóg 2. Cearnóg \sqrt{3}.

$$\left(2+\sqrt{3}\right)^{2}+\frac{\left(2-\sqrt{3}\right)\left(2-\sqrt{3}\right)}{4-3}+\frac{\sqrt{3}+1}{\sqrt{3}-1}$$

Dealaigh 3 ó 4 chun 1 a fháil.

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

Tugann aon rud a roinntear ar a haon é féin.

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

Méadaigh 2-\sqrt{3} agus 2-\sqrt{3} chun \left(2-\sqrt{3}\right)^{2} a fháil.

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

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi \sqrt{3}+1 chun ainmneoir \frac{\sqrt{3}+1}{\sqrt{3}-1} a thiontú in uimhir chóimheasta.

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

Mar shampla \left(\sqrt{3}-1\right)\left(\sqrt{3}+1\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}.

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

Cearnóg \sqrt{3}. Cearnóg 1.

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

Dealaigh 1 ó 3 chun 2 a fháil.

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

Méadaigh \sqrt{3}+1 agus \sqrt{3}+1 chun \left(\sqrt{3}+1\right)^{2} a fháil.

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

Úsáid an teoirim dhéthéarmach \left(a+b\right)^{2}=a^{2}+2ab+b^{2} chun \left(\sqrt{3}+1\right)^{2} a leathnú.

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

Is é 3 uimhir chearnach \sqrt{3}.

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

Suimigh 3 agus 1 chun 4 a fháil.

$$\left(2+\sqrt{3}\right)^{2}+\left(2-\sqrt{3}\right)^{2}+\frac{4+2\sqrt{3}}{2}$$

Roinn 4+2\sqrt{3} faoi 2 chun 2+\sqrt{3} a fháil.

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

Úsáid an teoirim dhéthéarmach \left(a+b\right)^{2}=a^{2}+2ab+b^{2} chun \left(2+\sqrt{3}\right)^{2} a leathnú.

$$4+4\sqrt{3}+\left(\sqrt{3}\right)^{2}+\left(2-\sqrt{3}\right)^{2}+2+\sqrt{3}$$

Is é 3 uimhir chearnach \sqrt{3}.

$$4+4\sqrt{3}+3+\left(2-\sqrt{3}\right)^{2}+2+\sqrt{3}$$

Suimigh 4 agus 3 chun 7 a fháil.

$$7+4\sqrt{3}+\left(2-\sqrt{3}\right)^{2}+2+\sqrt{3}$$

Úsáid an teoirim dhéthéarmach \left(a-b\right)^{2}=a^{2}-2ab+b^{2} chun \left(2-\sqrt{3}\right)^{2} a leathnú.

$$7+4\sqrt{3}+4-4\sqrt{3}+\left(\sqrt{3}\right)^{2}+2+\sqrt{3}$$

Is é 3 uimhir chearnach \sqrt{3}.

$$7+4\sqrt{3}+4-4\sqrt{3}+3+2+\sqrt{3}$$

Suimigh 4 agus 3 chun 7 a fháil.

$$7+4\sqrt{3}+7-4\sqrt{3}+2+\sqrt{3}$$

Suimigh 7 agus 7 chun 14 a fháil.

$$14+4\sqrt{3}-4\sqrt{3}+2+\sqrt{3}$$

Comhcheangail 4\sqrt{3} agus -4\sqrt{3} chun 0 a fháil.

$$14+2+\sqrt{3}$$

Suimigh 14 agus 2 chun 16 a fháil.

$$16+\sqrt{3}$$