Datrys frac{31sqrt{2}+31sqrt{5}}{2sqrt{10}-3}-frac{62sqrt{2}}{3-2sqrt{10}}

$$\frac { 31 \sqrt { 2 } + 31 \sqrt { 5 } } { 2 \sqrt { 10 } - 3 } - \frac { 62 \sqrt { 2 } } { 3 - 2 \sqrt { 10 } }$$

Enrhifo

$$15\sqrt{5}+19\sqrt{2}\approx 60.411077348$$

Camau Ateb

Mae'n rhesymoli enwadur \frac{31\sqrt{2}+31\sqrt{5}}{2\sqrt{10}-3} drwy luosi'r rhifiadur a'r enwadur â 2\sqrt{10}+3.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{\left(2\sqrt{10}-3\right)\left(2\sqrt{10}+3\right)}-\frac{62\sqrt{2}}{3-2\sqrt{10}}$$

Ystyriwch \left(2\sqrt{10}-3\right)\left(2\sqrt{10}+3\right). Gellir trawsnewid lluosi yn wahaniaeth rhwng sgwariau drwy ddefnyddio’r rheol: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{\left(2\sqrt{10}\right)^{2}-3^{2}}-\frac{62\sqrt{2}}{3-2\sqrt{10}}$$

Ehangu \left(2\sqrt{10}\right)^{2}.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{2^{2}\left(\sqrt{10}\right)^{2}-3^{2}}-\frac{62\sqrt{2}}{3-2\sqrt{10}}$$

Cyfrifo 2 i bŵer 2 a chael 4.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{4\left(\sqrt{10}\right)^{2}-3^{2}}-\frac{62\sqrt{2}}{3-2\sqrt{10}}$$

Sgwâr \sqrt{10} yw 10.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{4\times 10-3^{2}}-\frac{62\sqrt{2}}{3-2\sqrt{10}}$$

Lluosi 4 a 10 i gael 40.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{40-3^{2}}-\frac{62\sqrt{2}}{3-2\sqrt{10}}$$

Cyfrifo 3 i bŵer 2 a chael 9.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{40-9}-\frac{62\sqrt{2}}{3-2\sqrt{10}}$$

Tynnu 9 o 40 i gael 31.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}}{3-2\sqrt{10}}$$

Mae'n rhesymoli enwadur \frac{62\sqrt{2}}{3-2\sqrt{10}} drwy luosi'r rhifiadur a'r enwadur â 3+2\sqrt{10}.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}\left(3+2\sqrt{10}\right)}{\left(3-2\sqrt{10}\right)\left(3+2\sqrt{10}\right)}$$

Ystyriwch \left(3-2\sqrt{10}\right)\left(3+2\sqrt{10}\right). Gellir trawsnewid lluosi yn wahaniaeth rhwng sgwariau drwy ddefnyddio’r rheol: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}\left(3+2\sqrt{10}\right)}{3^{2}-\left(-2\sqrt{10}\right)^{2}}$$

Cyfrifo 3 i bŵer 2 a chael 9.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}\left(3+2\sqrt{10}\right)}{9-\left(-2\sqrt{10}\right)^{2}}$$

Ehangu \left(-2\sqrt{10}\right)^{2}.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}\left(3+2\sqrt{10}\right)}{9-\left(-2\right)^{2}\left(\sqrt{10}\right)^{2}}$$

Cyfrifo -2 i bŵer 2 a chael 4.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}\left(3+2\sqrt{10}\right)}{9-4\left(\sqrt{10}\right)^{2}}$$

Sgwâr \sqrt{10} yw 10.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}\left(3+2\sqrt{10}\right)}{9-4\times 10}$$

Lluosi 4 a 10 i gael 40.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}\left(3+2\sqrt{10}\right)}{9-40}$$

Tynnu 40 o 9 i gael -31.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}\left(3+2\sqrt{10}\right)}{-31}$$

Rhannu 62\sqrt{2}\left(3+2\sqrt{10}\right) â -31 i gael -2\sqrt{2}\left(3+2\sqrt{10}\right).

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\left(-2\sqrt{2}\left(3+2\sqrt{10}\right)\right)$$

Gwrthwyneb -2\sqrt{2}\left(3+2\sqrt{10}\right) yw 2\sqrt{2}\left(3+2\sqrt{10}\right).

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Cyfrifwch y briodoledd ddosrannol drwy luosi pob 31\sqrt{2}+31\sqrt{5} gan bob 2\sqrt{10}+3.

$$\frac{62\sqrt{10}\sqrt{2}+93\sqrt{2}+62\sqrt{5}\sqrt{10}+93\sqrt{5}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Ffactora 10=2\times 5. Ailysgrifennu ail isradd y lluoswm \sqrt{2\times 5} fel lluoswm ail israddau \sqrt{2}\sqrt{5}.

$$\frac{62\sqrt{2}\sqrt{5}\sqrt{2}+93\sqrt{2}+62\sqrt{5}\sqrt{10}+93\sqrt{5}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Lluosi \sqrt{2} a \sqrt{2} i gael 2.

$$\frac{62\times 2\sqrt{5}+93\sqrt{2}+62\sqrt{5}\sqrt{10}+93\sqrt{5}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Lluosi 62 a 2 i gael 124.

$$\frac{124\sqrt{5}+93\sqrt{2}+62\sqrt{5}\sqrt{10}+93\sqrt{5}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Ffactora 10=5\times 2. Ailysgrifennu ail isradd y lluoswm \sqrt{5\times 2} fel lluoswm ail israddau \sqrt{5}\sqrt{2}.

$$\frac{124\sqrt{5}+93\sqrt{2}+62\sqrt{5}\sqrt{5}\sqrt{2}+93\sqrt{5}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Lluosi \sqrt{5} a \sqrt{5} i gael 5.

$$\frac{124\sqrt{5}+93\sqrt{2}+62\times 5\sqrt{2}+93\sqrt{5}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Lluosi 62 a 5 i gael 310.

$$\frac{124\sqrt{5}+93\sqrt{2}+310\sqrt{2}+93\sqrt{5}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Cyfuno 93\sqrt{2} a 310\sqrt{2} i gael 403\sqrt{2}.

$$\frac{124\sqrt{5}+403\sqrt{2}+93\sqrt{5}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Cyfuno 124\sqrt{5} a 93\sqrt{5} i gael 217\sqrt{5}.

$$\frac{217\sqrt{5}+403\sqrt{2}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Rhannu pob term 217\sqrt{5}+403\sqrt{2} â 31 i gael 7\sqrt{5}+13\sqrt{2}.

$$7\sqrt{5}+13\sqrt{2}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Defnyddio’r briodwedd ddosbarthu i luosi 2\sqrt{2} â 3+2\sqrt{10}.

$$7\sqrt{5}+13\sqrt{2}+6\sqrt{2}+4\sqrt{10}\sqrt{2}$$

Ffactora 10=2\times 5. Ailysgrifennu ail isradd y lluoswm \sqrt{2\times 5} fel lluoswm ail israddau \sqrt{2}\sqrt{5}.

$$7\sqrt{5}+13\sqrt{2}+6\sqrt{2}+4\sqrt{2}\sqrt{5}\sqrt{2}$$

Lluosi \sqrt{2} a \sqrt{2} i gael 2.

$$7\sqrt{5}+13\sqrt{2}+6\sqrt{2}+4\times 2\sqrt{5}$$

Lluosi 4 a 2 i gael 8.

$$7\sqrt{5}+13\sqrt{2}+6\sqrt{2}+8\sqrt{5}$$

Cyfuno 13\sqrt{2} a 6\sqrt{2} i gael 19\sqrt{2}.

$$7\sqrt{5}+19\sqrt{2}+8\sqrt{5}$$

Cyfuno 7\sqrt{5} a 8\sqrt{5} i gael 15\sqrt{5}.

$$15\sqrt{5}+19\sqrt{2}$$

Ffactor

$$15 \sqrt{5} + 19 \sqrt{2} = 60.411077348$$