Datrys (sqrt{2}+sqrt{5})^2-(2+sqrt{10})^2+sqrt{90}+(2sqrt{2}-1)(2sqrt{2}+1)

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

Enrhifo

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

Camau Ateb

Defnyddio'r theorem binomaidd \left(a+b\right)^{2}=a^{2}+2ab+b^{2} i ehangu'r \left(\sqrt{2}+\sqrt{5}\right)^{2}.

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

Sgwâr \sqrt{2} yw 2.

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

I luosi \sqrt{2} a \sqrt{5}, dylid lluosi'r rhifau dan yr ail isradd.

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

Sgwâr \sqrt{5} yw 5.

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

Adio 2 a 5 i gael 7.

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

Defnyddio'r theorem binomaidd \left(a+b\right)^{2}=a^{2}+2ab+b^{2} i ehangu'r \left(2+\sqrt{10}\right)^{2}.

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

Sgwâr \sqrt{10} yw 10.

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

Adio 4 a 10 i gael 14.

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

I ddod o hyd i wrthwyneb 14+4\sqrt{10}, dewch o hyd i wrthwyneb pob term.

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

Tynnu 14 o 7 i gael -7.

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

Cyfuno 2\sqrt{10} a -4\sqrt{10} i gael -2\sqrt{10}.

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

Ffactora 90=3^{2}\times 10. Ailysgrifennu ail isradd y lluoswm \sqrt{3^{2}\times 10} fel lluoswm ail israddau \sqrt{3^{2}}\sqrt{10}. Cymryd isradd 3^{2}.

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

Cyfuno -2\sqrt{10} a 3\sqrt{10} i gael \sqrt{10}.

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

Ystyriwch \left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right). Gellir trawsnewid lluosi yn wahaniaeth rhwng sgwariau drwy ddefnyddio’r rheol: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Sgwâr 1.

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

Ehangu \left(2\sqrt{2}\right)^{2}.

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

Cyfrifo 2 i bŵer 2 a chael 4.

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

Sgwâr \sqrt{2} yw 2.

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

Lluosi 4 a 2 i gael 8.

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

Tynnu 1 o 8 i gael 7.

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

Adio -7 a 7 i gael 0.

$$\sqrt{10}$$