Réitigh {left(sqrt{2}+sqrt{5}right)}^2-{left(2+sqrt{10}right)}^2+sqrt{90}+left(2sqrt{2}-1right)left(2sqrt{2}+1right)

$${ \left( \sqrt{ 2 } + \sqrt{ 5 } \right) }^{ 2 } - { \left(2+ \sqrt{ 10 } \right) }^{ 2 } + \sqrt{ 90 } + \left( 2 \sqrt{ 2 } -1 \right) \left( 2 \sqrt{ 2 } +1 \right)$$

Luacháil

$$\sqrt{10}\approx 3.16227766$$

Céimeanna Réitigh

Úsáid an teoirim dhéthéarmach \left(a+b\right)^{2}=a^{2}+2ab+b^{2} chun \left(\sqrt{2}+\sqrt{5}\right)^{2} a leathnú.

$$\left(\sqrt{2}\right)^{2}+2\sqrt{2}\sqrt{5}+\left(\sqrt{5}\right)^{2}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Is é 2 uimhir chearnach \sqrt{2}.

$$2+2\sqrt{2}\sqrt{5}+\left(\sqrt{5}\right)^{2}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Iolraigh na huimhreacha faoin bhfréamh cearnach chun \sqrt{2} agus \sqrt{5} a iolrú.

$$2+2\sqrt{10}+\left(\sqrt{5}\right)^{2}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Is é 5 uimhir chearnach \sqrt{5}.

$$2+2\sqrt{10}+5-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Suimigh 2 agus 5 chun 7 a fháil.

$$7+2\sqrt{10}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Úsáid an teoirim dhéthéarmach \left(a+b\right)^{2}=a^{2}+2ab+b^{2} chun \left(2+\sqrt{10}\right)^{2} a leathnú.

$$7+2\sqrt{10}-\left(4+4\sqrt{10}+\left(\sqrt{10}\right)^{2}\right)+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Is é 10 uimhir chearnach \sqrt{10}.

$$7+2\sqrt{10}-\left(4+4\sqrt{10}+10\right)+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Suimigh 4 agus 10 chun 14 a fháil.

$$7+2\sqrt{10}-\left(14+4\sqrt{10}\right)+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Chun an mhalairt ar 14+4\sqrt{10} a aimsiú, aimsigh an mhalairt ar gach téarma.

$$7+2\sqrt{10}-14-4\sqrt{10}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Dealaigh 14 ó 7 chun -7 a fháil.

$$-7+2\sqrt{10}-4\sqrt{10}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Comhcheangail 2\sqrt{10} agus -4\sqrt{10} chun -2\sqrt{10} a fháil.

$$-7-2\sqrt{10}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Fachtóirigh 90=3^{2}\times 10. Athscríobh fréamh cearnach an toraidh \sqrt{3^{2}\times 10} mar thoradh na bhfréamhacha cearnacha \sqrt{3^{2}}\sqrt{10}. Tóg fréamh chearnach 3^{2}.

$$-7-2\sqrt{10}+3\sqrt{10}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Comhcheangail -2\sqrt{10} agus 3\sqrt{10} chun \sqrt{10} a fháil.

$$-7+\sqrt{10}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Mar shampla \left(2\sqrt{2}-1\right)\left(2\sqrt{2}+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}. Cearnóg 1.

$$-7+\sqrt{10}+\left(2\sqrt{2}\right)^{2}-1$$

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

$$-7+\sqrt{10}+2^{2}\left(\sqrt{2}\right)^{2}-1$$

Ríomh cumhacht 2 de 2 agus faigh 4.

$$-7+\sqrt{10}+4\left(\sqrt{2}\right)^{2}-1$$

Is é 2 uimhir chearnach \sqrt{2}.

$$-7+\sqrt{10}+4\times 2-1$$

Méadaigh 4 agus 2 chun 8 a fháil.

$$-7+\sqrt{10}+8-1$$

Dealaigh 1 ó 8 chun 7 a fháil.

$$-7+\sqrt{10}+7$$

Suimigh -7 agus 7 chun 0 a fháil.

$$\sqrt{10}$$